# Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board chapter 3 - Trigonometry - 2 [Latest edition]

## Chapter 3: Trigonometry - 2

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.5Miscellaneous Exercise 3
Exercise 3.1 [Pages 39 - 40]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 3 Trigonometry - 2 Exercise 3.1 [Pages 39 - 40]

Exercise 3.1 | Q 1. (i) | Page 39

Find the values of:

sin 15°

Exercise 3.1 | Q 1. (ii) | Page 39

Find the values of:

cos 75°

Exercise 3.1 | Q 1. (iii) | Page 39

Find the values of:

tan 105°

Exercise 3.1 | Q 1. (iv) | Page 39

Find the values of:

cot 225°

Exercise 3.1 | Q 2. (i) | Page 39

Prove the following:

cos (pi/2 - x) cos(pi/2 - y) - sin(pi/2 - x) sin(pi/2 - y) = – cos (x + y)

Exercise 3.1 | Q 2. (ii) | Page 39

Prove the following:

tan(pi/4 + theta) = (1 - tan theta)/(1 + tan theta)

Exercise 3.1 | Q 2. (iii) | Page 39

Prove the following:

((1 + tan x)/(1 - tan x))^2 = tan(pi/4 + x)/(tan(pi/4 - x))

Exercise 3.1 | Q 2. (iv) | Page 39

Prove the following:

sin [(n + 1)A]. sin [(n + 2)A] + cos [(n + 1)A]. cos [(n + 2)A] = cos A

Exercise 3.1 | Q 2. (v) | Page 39

Prove the following:

sqrt(2)cos (pi/4 - "A") = cos A + sin A

Exercise 3.1 | Q 2. (vi) | Page 39

Prove the following:

(cos(x - y))/(cos(x + y)) = (cotx coty + 1)/(cotx coty - 1)

Exercise 3.1 | Q 2. (vii) | Page 39

Prove the following:

cos(x + y).cos(x − y) = cos2y − sin2x

Exercise 3.1 | Q 2. (viii) | Page 39

Prove the following:

(tan5"A" - tan3"A")/(tan5"A" + tan3"A") = (sin2"A")/(sin8"A")

Exercise 3.1 | Q 2. (ix) | Page 39

Prove the following:

tan8θ − tan5θ − tan3θ = tan8θ tan5θ tan3θ

Exercise 3.1 | Q 2. (x) | Page 39

Prove the following:

tan50° = tan40° + 2 tan10°

Exercise 3.1 | Q 2. (xi) | Page 39

Prove the following:

(cos27^circ + sin27^circ)/(cos27^circ - sin27^circ) = tan72°

Exercise 3.1 | Q 2. (xii) | Page 39

Prove the following:

tan10° + tan35° + tan10°.tan35° = 1

Exercise 3.1 | Q 2. (xiii) | Page 39

Prove the following:

(cot"A"cot4"A" + 1)/(cot"A" cot4"A" - 1) = (cos3"A")/(cos5"A")

Exercise 3.1 | Q 2. (xiv) | Page 39

Prove the following:

(cos15^circ - sin15^circ)/(cos15^circ + sin15^circ) = 1/sqrt(3)

Exercise 3.1 | Q 3. (i) | Page 40

If sin A = (-5)/13, pi < "A" < (3pi)/2 and cos B = 3/5, (3pi)/2 < "B" < 2pi find sin (A + B)

Exercise 3.1 | Q 3. (ii) | Page 40

If sin A = (-5)/13, pi < "A" < (3pi)/2 and cos B = 3/5, (3pi)/2 < "B" < 2pi find cos (A – B)

Exercise 3.1 | Q 3. (iii) | Page 40

If sin A = (-5)/13, pi < "A" < (3pi)/2 and cos B = 3/5, (3pi)/2 < "B" < 2pi find tan (A + B)

Exercise 3.1 | Q 4 | Page 40

If tan A = 5/6, tan "B" = 1/11, prove that A + B = pi/4

Exercise 3.2 [Page 42]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 3 Trigonometry - 2 Exercise 3.2 [Page 42]

Exercise 3.2 | Q 1. (i) | Page 42

Find the value of :

sin 690°

Exercise 3.2 | Q 1. (ii) | Page 42

Find the value of :

sin (495°)

Exercise 3.2 | Q 1. (iii) | Page 42

Find the value of :

cos 315°

Exercise 3.2 | Q 1. (iv) | Page 42

Find the value of :

cos (600°)

Exercise 3.2 | Q 1. (v) | Page 42

Find the value of :

tan 225°

Exercise 3.2 | Q 1. (vi) | Page 42

Find the value of :

tan (– 690°)

Exercise 3.2 | Q 1. (vii) | Page 42

Find the value of :

sec 240°

Exercise 3.2 | Q 1. (viii) | Page 42

Find the value of :

sec (– 855°)

Exercise 3.2 | Q 1. (ix) | Page 42

Find the value of :

cosec 780°

Exercise 3.2 | Q 1. (x) | Page 42

Find the value of :

cot (– 1110°)

Exercise 3.2 | Q 2. (i) | Page 42

Prove the following:

(cos(pi + x) cos(-x))/(sin(pi - x)cos(pi/2 + x)) = cot2x

Exercise 3.2 | Q 2. (ii) | Page 42

Prove the following:

cos((3pi)/2 + x) cos(2pi + x)[cot((3pi)/2 - x) + cot(2pi + x)] = 1

Exercise 3.2 | Q 2. (iii) | Page 42

Prove the following:

sec 840° . cot (– 945°) + sin 600° tan (– 690°) = 3/2

Exercise 3.2 | Q 2. (iv) | Page 42

Prove the following:

("cosec"(90^circ - x)sin(180^circ - x)cot(360^circ - x))/(sec(180^circ + x)tan(90^circ + x)sin(-x)) = 1

Exercise 3.2 | Q 2. (v) | Page 42

Prove the following:

(sin^3(pi + x)sec^2(pi - x)tan(2pi - x))/(cos^2(pi/2 + x)sin(pi - x)"cosec"^2 - x) = tan3x

Exercise 3.2 | Q 2. (vi) | Page 42

Prove the following:

cosθ + sin (270° + θ) − sin (270° − θ) + cos (180° + θ) = 0

Exercise 3.3 [Page 48]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 3 Trigonometry - 2 Exercise 3.3 [Page 48]

Exercise 3.3 | Q 1. (i) | Page 48

Find the value of :

sin  pi/8

Exercise 3.3 | Q 1. (ii) | Page 48

Find the value of :

cos  pi/8

Exercise 3.3 | Q 2 | Page 48

Find sin 2x, cos 2x, tan 2x if secx = (-13)/5, pi/2 < x < pi

Exercise 3.3 | Q 3. (i) | Page 48

Prove the following:

(1 - cos2theta)/(1 + cos2theta) = tan2θ

Exercise 3.3 | Q 3. (ii) | Page 48

Prove the following:

(sin 3x + sin x)sin x + (cos 3x – cos x) cos x = 0

Exercise 3.3 | Q 3. (iii) | Page 48

Prove the following:

(cos x + cos y)2 + (sin x – sin y)2 = 4cos^2  ((x + y))/2

Exercise 3.3 | Q 3. (iv) | Page 48

Prove the following:

(cos x – cos y)2 + (sin x – sin y)2 = 4sin^2  ((x - y))/2

Exercise 3.3 | Q 3. (v) | Page 48

Prove the following:

tan x + cot x = 2 cosec 2x

Exercise 3.3 | Q 3. (vi) | Page 48

Prove the following:

(cosx + sinx)/(cosx - sinx) - (cosx - sinx)/(cosx + sinx) = 2tan2x

Exercise 3.3 | Q 3. (vii) | Page 48

Prove the following:

sqrt(2 + sqrt(2 + sqrt(2 + 2cos8x) = 2 cos x

Exercise 3.3 | Q 3. (viii) | Page 48

Prove the following:

16 sin θ cos θ cos 2θ cos 4θ cos 8θ = sin 16θ

Exercise 3.3 | Q 3. (ix) | Page 48

Prove the following:

(sin3x)/cosx + (cos3x)/sinx = 2 cot 2x

Exercise 3.3 | Q 3. (x) | Page 48

Prove the following:

cosx/(1 + sinx) = (cot(x/2) - 1)/(cot(x/2) + 1)

Exercise 3.3 | Q 3. (xi) | Page 48

Prove the following:

(tan(theta/2) + cot(theta/2))/(cot(theta/2) - tan(theta/2)) = secθ

Exercise 3.3 | Q 3. (xii) | Page 48

Prove the following:

1/(tan3"A" - tan"A") - 1/(cot3"A" - cot"A") = cot2A

Exercise 3.3 | Q 3. (xiii) | Page 48

Prove the following:

cos7° cos 14° cos28° cos 56° = sin68^circ/(16cos83^circ)

Exercise 3.3 | Q 3. (xiv) | Page 48

Prove the following:

(sin^2(-160^circ))/(sin^(2)70^circ) + sin(180^circ - theta)/sintheta = sec220°

Exercise 3.3 | Q 3. (xv) | Page 48

Prove the following:

(2cos 4x + 1)/(2cosx + 1) = (2 cos x – 1) (2 cos 2x – 1)

Exercise 3.3 | Q 3. (xvi) | Page 48

Prove the following:

cos2x + cos2(x + 120°) + cos2(x – 120°) = 3/2

Exercise 3.3 | Q 3. (xvii) | Page 48

Prove the following:

2cosec 2x + cosec x = secx cot(x/2)

Exercise 3.3 | Q 3. (xviii) | Page 48

Prove the following:

4 cos x. cos(x + pi/3) . cos(x - pi/3) = cos 3x

Exercise 3.3 | Q 3. (xix) | Page 48

Prove the following:

sinx tan(x/2) + 2cosx = 2/(1 + tan^2(x/2))

Exercise 3.4 [Page 51]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 3 Trigonometry - 2 Exercise 3.4 [Page 51]

Exercise 3.4 | Q 1. (i) | Page 51

Express the following as a sum or difference of two trigonometric function:

2 sin 4x cos 2x

Exercise 3.4 | Q 1. (ii) | Page 51

Express the following as a sum or difference of two trigonometric function:

2sin  (2pi)/3 cos  pi/2

Exercise 3.4 | Q 1. (iii) | Page 51

Express the following as a sum or difference of two trigonometric function:

2 cos 4θ cos 2θ

Exercise 3.4 | Q 1. (iv) | Page 51

Express the following as a sum or difference of two trigonometric function:

2 cos 35° cos 75°

Exercise 3.4 | Q 2. (i) | Page 51

Prove the following :

(sin2x + sin2y)/(sin2x - sin2y) = (tan(x + y))/(tan(x - y))

Exercise 3.4 | Q 2. (ii) | Page 51

Prove the following :

sin 6x + sin 4x – sin 2x = 4 cos x sin 2x cos 3x

Exercise 3.4 | Q 2. (iii) | Page 51

Prove the following :

(sinx - sin3x + sin5x - sin7x)/(cosx - cos3x - cos5x + cos7x) = cot2x

Exercise 3.4 | Q 2. (iv) | Page 51

Prove the following :

sin 18° cos 39° + sin 6° cos 15° = sin 24° cos 33°

Exercise 3.4 | Q 2. (v) | Page 51

Prove the following :

cos 20° cos 40° cos 60° cos 80° = 1/16

Exercise 3.4 | Q 2. (vi) | Page 51

Prove the following :

sin 20° sin 40° sin 60° sin 80° = 3/16

Exercise 3.5 [Page 54]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 3 Trigonometry - 2 Exercise 3.5 [Page 54]

Exercise 3.5 | Q 1 | Page 54

In ΔABC, A + B + C = π show that

cos 2A + cos 2B + cos 2C = –1 – 4 cos A cos B cos C

Exercise 3.5 | Q 2 | Page 54

In ΔABC, A + B + C = π show that

sin A + sin B + sin C = 4cos  "A"/2  cos  "B"/2  cos  "C"/2

Exercise 3.5 | Q 3 | Page 54

In ΔABC, A + B + C = π show that

cos A + cos B – cos C = 4cos  "A"/2  cos  "B"/2  sin  "C"/2 - 1

Exercise 3.5 | Q 4 | Page 54

In ΔABC, A + B + C = π show that

sin2A + sin2B − sin2C = 2 sin A sin B cos C

Exercise 3.5 | Q 5 | Page 54

In ΔABC, A + B + C = π show that

sin^2  "A"/2 + sin^2  "B"/2 - sin^2  "C"/2 = 1 - 2cos  "A"/2  cos  "B"/2 sin  "C"/2

Exercise 3.5 | Q 6 | Page 54

In ΔABC, A + B + C = π show that

tan  "A"/2 tan  "B"/2 + tan  "B"/2 tan  "C"/2 + tan  "C"/2tan  "A"/2 = 1

Exercise 3.5 | Q 7 | Page 54

In ΔABC, A + B + C = π show that

cot  "A"/2 + cot  "B"/2 + cot  "C"/2 = cot  "A"/2  cot  "B"/2 cot  "C"/2

Exercise 3.5 | Q 8 | Page 54

In ΔABC, A + B + C = π show that

tan 2A + tan 2B + tan 2C = tan 2A tan 2B tan 2C

Exercise 3.5 | Q 9 | Page 54

In ΔABC, A + B + C = π show that

cos2A +cos2B – cos2C = 1 – 2 sin A sin B cos C

Miscellaneous Exercise 3 [Pages 56 - 58]

### Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board Chapter 3 Trigonometry - 2 Miscellaneous Exercise 3 [Pages 56 - 58]

Miscellaneous Exercise 3 | Q I. (1) | Page 56

Select the correct option from the given alternatives :

The value of sin(n + 1) A sin (n + 2) A + cos(n + 1) A cos(n + 2) A is equal to

• sin A

• cos A

• – cos A

• sin 2A

Miscellaneous Exercise 3 | Q I. (2) | Page 56

Select the correct option from the given alternatives :

If tan A – tan B = x and cot B – cot A = y, then cot (A – B) = _____

• 1/y - 1/x

• 1/x - 1/y

• 1/x + 1/y

• (xy)/(x - y)

Miscellaneous Exercise 3 | Q I. (3) | Page 56

Select the correct option from the given alternatives :

If sin θ = n sin (θ + 2α), then tan (θ + α) is equal to

• (1 + n)/(2 - n) tan alpha

• (1 - n)/(1 + n) tan alpha

• tan α

• (1 + n)/(1 - n) tan alpha

Miscellaneous Exercise 3 | Q I. (iv) | Page 57

Select the correct option from the given alternatives :

The value of costheta/(1 + sin theta) is equal to ……

• tan(theta/2 - pi/4)

• tan(-pi/4 - theta/2)

• tan(pi/4 - theta/2)

• tan(pi/4 + theta/2)

Miscellaneous Exercise 3 | Q I. (v) | Page 57

Select the correct option from the given alternatives :

The value of cos A cos (60° – A) cos (60° + A) is equal to ......

• 1/2cos3"A"

• cos 3A

• 1/4cos3"A"

• 4 cos 3A

Miscellaneous Exercise 3 | Q I. (vi) | Page 57

Select the correct option from the given alternatives :

The value of sin  pi/14sin  (3pi)/14sin  (5pi)/14sin  (7pi)/14sin  (9pi)/14sin  (11pi)/14sin  (13pi)/14 is ....

• 1/16

• 1/64

• 1/128

• 1/256

Miscellaneous Exercise 3 | Q I. (vii) | Page 57

Select the correct option from the given alternatives :

If α + β + γ = π then the value of sin2α + sin2β – sin2γ is equal to …......

• 2 sin α

• 2 sin α cos β sin γ

• 2 sin α sin β cos γ

• 2 sin α sin β sin γ

Miscellaneous Exercise 3 | Q I. (viii) | Page 57

Select the correct option from the given alternatives :

Let 0 < A, B < pi/2 satisfying the equation 3 sin2A + 2 sin2B = 1 and 3 sin 2A − 2 sin 2B = 0 then A + 2B is equal to ______

• π

• pi/2

• pi/4

Miscellaneous Exercise 3 | Q I, (ix) | Page 57

Select the correct option from the given alternatives :

In ∆ABC if cot A cot B cot C > 0 then the triangle is _________

• Acute angled

• right angled

• obtuse angled

• isosceles right angled

Miscellaneous Exercise 3 | Q I. (x) | Page 57

Select the correct option from the given alternatives :

The numerical value of tan 20° tan 80° cot 50° is equal to ______.

• sqrt(3)

• 1/sqrt(3)

• 2sqrt(3)

• 1/(2sqrt(3)

Miscellaneous Exercise 3 | Q II. (1) | Page 57

Prove the following:

tan 20° tan 80° cot 50° = sqrt(3)

Miscellaneous Exercise 3 | Q II. (2) | Page 57

Prove the following:

If sin α sin β − cos α cos β + 1 = 0 then prove cot α tan β = −1

Miscellaneous Exercise 3 | Q II. (3) | Page 57

Prove the following:

cos  (2pi)/15 cos  (4pi)/15cos  (8pi)/15cos  (16pi)/15 = 1/16

Miscellaneous Exercise 3 | Q II. (4) | Page 57

Prove the following:

(1 + cos  pi/8)(1 + cos  (3pi)/8)(1 + cos  (5pi)/8)(1 + cos  (7pi)/8) = 1/8

Miscellaneous Exercise 3 | Q II. (5) | Page 57

Prove the following:

cos 12°+ cos 84° + cos 156° + cos 132° = -1/2

Miscellaneous Exercise 3 | Q II. (6) | Page 57

Prove the following:

cos(pi/4 + x) + cos(pi/4 - x) = sqrt(2)cosx

Miscellaneous Exercise 3 | Q II. (7) | Page 57

Prove the following:

(sin5x - 2sin3x + sinx)/(cos5x - cosx) = tanx

Miscellaneous Exercise 3 | Q II. (8) | Page 57

Prove the following:

sin26x − sin24x = sin2x sin10x

Miscellaneous Exercise 3 | Q II . (9) | Page 57

Prove the following:

cos22x − cos26x = sin4x sin8x

Miscellaneous Exercise 3 | Q II. (10) | Page 57

Prove the following:

cot4x (sin5x + sin3x) = cotx (sin5x − sin3x)

Miscellaneous Exercise 3 | Q II. (11) | Page 57

Prove the following:

(cos9x - cos5x)/(sin17x - sin3x) = - (sin2x)/(cos10x)

Miscellaneous Exercise 3 | Q II. (12) | Page 57

Prove the following:

If sin 2A = λsin 2B then prove that (tan("A" + "B"))/(tan("A" - "B")) = (lambda + 1)/(lambda - 1)

Miscellaneous Exercise 3 | Q II. (13) | Page 57

Prove the following:

(2cos2"A" + 1)/(2cos2"A" - 1) = tan(60° + A) tan(60° − A)

Miscellaneous Exercise 3 | Q II. (14) | Page 57

Prove the following:

tanA + tan(60° + A) + tan(120° + A) = 3 tan 3A

Miscellaneous Exercise 3 | Q II. (15) | Page 57

Prove the following:

3tan610° – 27 tan410° + 33tan210° = 1

Miscellaneous Exercise 3 | Q II. (16) | Page 58

Prove the following:

cosec 48° + cosec 96° + cosec 192° + cosec 384° = 0

Miscellaneous Exercise 3 | Q II. (17) | Page 58

Prove the following:

3(sin x – cos x)4 + 6(sin x + cos x)2 + 4(sin6x + cos6x) = 13

Miscellaneous Exercise 3 | Q II. (18) | Page 58

Prove the following:

tan A + 2 tan 2A + 4 tan 4A + 8 cot 8A = cot A

Miscellaneous Exercise 3 | Q II. (19) | Page 58

Prove the following:

If A + B + C = (3pi)/2, then cos 2A + cos 2B + cos 2C = 1 − 4 sin A sin B sin C

Miscellaneous Exercise 3 | Q II. (20) | Page 58

Prove the following:

In any triangle ABC, sin A − cos B = cos C then ∠B = pi/2.

Miscellaneous Exercise 3 | Q II. (21) | Page 58

Prove the following:

tan^3x/(1 + tan^2x) + cot^3x/(1 + cot^2x) = secx cosecx − 2sinx cosx

Miscellaneous Exercise 3 | Q II. (22) | Page 58

Prove the following:

sin 20° sin 40° sin 80° = sqrt(3)/8

Miscellaneous Exercise 3 | Q II. (23) | Page 58

Prove the following:

sin 18° = (sqrt(5) - 1)/4

Miscellaneous Exercise 3 | Q II. (24) | Page 58

Prove the following:

cos 36° = (sqrt(5) + 1)/4

Miscellaneous Exercise 3 | Q II. (25) | Page 58

Prove the following:

sin 36° = (sqrt(10 - 2sqrt(5)))/4

Miscellaneous Exercise 3 | Q II. (26) | Page 58

Prove the following:

sin  pi^"c"/8 = 1/2sqrt(2 - sqrt(2))

Miscellaneous Exercise 3 | Q II. (27) | Page 58

Prove the following:

tan  pi/8 = sqrt(2) - 1

Miscellaneous Exercise 3 | Q II. (28) | Page 58

Prove the following:

tan6° tan42° tan66° tan78° = 1

Miscellaneous Exercise 3 | Q II. (29) | Page 58

Prove the following:

sin47° + sin61° − sin11° − sin25° = cos7°

Miscellaneous Exercise 3 | Q II. (30) | Page 58

Prove the following:

sqrt(3)  "cosec"20^circ - sec20^circ = 4

Miscellaneous Exercise 3 | Q II. (31) | Page 58

Prove the following:

In ∆ABC, ∠C = (2pi)/3, then prove that cos2A + cos2B − cos A cos B = 3/4

## Chapter 3: Trigonometry - 2

Exercise 3.1Exercise 3.2Exercise 3.3Exercise 3.4Exercise 3.5Miscellaneous Exercise 3

## Balbharati solutions for Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board chapter 3 - Trigonometry - 2

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Concepts covered in Mathematics and Statistics 1 (Arts and Science) 11th Standard Maharashtra State Board chapter 3 Trigonometry - 2 are Trigonometric Functions of Sum and Difference of Angles, Trigonometric Functions of Allied Angels, Trigonometric Functions of Multiple Angles, Trigonometric Functions of Double Angles, Trigonometric Functions of Triple Angle, Factorization Formulae, Formulae for Conversion of Sum Or Difference into Product, Formulae for Conversion of Product in to Sum Or Difference, Trigonometric Functions of Angles of a Triangle.

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