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# Selina solutions for Class 10 Mathematics chapter 21 - Trigonometrical Identities

## Selina ICSE Concise Mathematics for Class 10 (2018-2019)

#### Selina Selina ICSE Concise Mathematics Class 10 (2018-2019) ## Chapter 21: Trigonometrical Identities

21A21B21C21D21E21E 21E

#### Chapter 21: Trigonometrical Identities Exercise 21A solutions [Page 0]

Prove.
(secA-1)/(secA+1)=(1-cosA)/(1+cosA)

Prove.
(1+sinA)/(1-sinA)=(co   secA+1)/(co   sinA-1

Prove.
1/(tanA+cotA)=cosAsinA

Prove.
tanA-cotA=(1-2cos^2A)/(sinAcosA)

Prove.
sin^4A-cos^4A=2sin^2A-1

Prove.
(1-tanA)^2+(1+tanA)^2=2sec^2A

Prove.
cosecA - cosec2 A = cot4 A + cot2 A

Prove.
sec A (1-sin A) (sec A + tan A) = 1

Prove.
cosec A(1+ cos A) (cosecA - cot A) =1

Prove.
sec2 A + cosec2 A = sec2 A cosec2 A

Prove.
((1+tan^2A)cotA)/(cosec^2A)=tanA

Prove.
tan2A - sin2A = tan2A sin2A

Prove.
cot2 A - cos2 A = cos2 A.cot2 A

Prove.
(cosec A + sin A) (cosec A - sin A) = cot2 A + cos2

Prove.
(sec A - cos A) (sec A + cos A) = sin2 A + tan2

Prove.
(cosA + sinA)2 + (cosA - sinA)2 = 2

Prove.
(cosec A - sin A) (sec A - cos A) (tan A + cot A) = 1

Prove.
1/(sec A+tanA)=secA-tanA

Prove.
cosecA+cotA=1/(cosecA-cotA)

Prove.
(secA-tanA)/(secA+tanA)=1-2secAtanA+2tan^2A

prove.
(sinA + cosecA)2 + (cosA + secA)2 = 7 + tan2A + cot2A

prove.
sec2A cosec2A = tan2A + cot2A + 2

Prove.
1/(1+cosA)+1/(1-cosA)=2cosec^2A

Prove.
1/(1-sinA)+1/(1+sinA)=2sec^2A

Prove.
(cosecA)/(cosecA-1)+(cosecA)/(cosecA+1)=2sec^2A

prove.
secA/(secA+1)+secA/(secA-1)=2cosec^2A

Prove.
(1+cosA)/(1-cosA)=tan^2A/(secA-1)^2

Prove.
cot^2A/(cosecA+1)^2=(1-sinA)/(1+sinA)

Prove.
(1+sinA)/cosA+cosA/(1+sinA)=2secA

Prove.
(1-sinA)/(1+sinA)=(secA-tanA)^2

Prove.
(cotA-cosecA)^2=(1-cosA)/(1+cosA)

Prove.
(cosecA-1)/(cosecA+1)=(cosA/(1+sinA))^2

Prove.
tan^2A-tan^2B=(sin^2A-sinB)/(cos^2Acos^2B

Prove.
(sinA-2sin^3A)/(2cos^3A-cosA)=tanA

Prove.
sinA/(1+cosA)=cosecA-cotA

Prove.
cosA/(1-sinA)=secA+tanA

Prove.
(sinAtanA)/(1-cosA)=1+secA

Prove.
(1 + cot A - cosec A)(1+ tan A + sec A) = 2

Prove.
sqrt((1+sinA)/(1-sinA))=secA+tanA

Prove.
sqrt((1-cosA)/(1+cosA))=cosecA-cotA

Prove.
sqrt((1-cosA)/(1+cosA))=sinA/((1+cosA)

Prove.
sqrt((1-sinA)/(1+sinA))=cosA/(1+sinA)

Prove.
1-cos^2A/(1+sinA)=sinA

Prove.
1/(sinA+cosA)+1/(sinA-cosA)=(2sinA)/(1-2cos^2A)

Prove.
(sinA+cosA)/(cosA-cosA)+(sinA-cosA)/(sinA+cosA)=2/(2sin^2A-1)

Prove.
(cotA+cosecA-1)/(cotA-cosecA+1)=(1+cosA)/sinA

Prove.
(sinthetatantheta)/(1-costheta)=1+sectheta

Prove.
(costhetacottheta)/(1+sintheta)=cosectheta-1

#### Chapter 21: Trigonometrical Identities Exercise 21B solutions [Page 0]

Prove.
cosA/(1-tanA)+sinA/(1-cotA)=sinA+cosA

Prove.
(cos^3A+sin^3A)/(cos^3A+sin^3A)+(cos^3A-sin^3A)/(cos^3A-sin^3A)=2

Prove.
tanA/(1-cotA)+cot/(1-tanA)=secA cosecA+1

Prove.
(tanA+1/cosA)^2+(tanA-1/cosA)^2=2((1+sin^2A)/(1-sin^2A))

Prove.
2 sin2A + cos4A = 1 + sin4

Prove.
(sinA-sinB)/(cosA+cosB)+(cosA-cosB)/(sinA+sinB)=0

Prove.
(cosecA-sinA)(secA-cosA)=1/(tanA+cotA)

Prove.
(1 + tanA tanB)2 + (tanA - tanB)2 = sec2A sec2

Prove.
1/(cosA+sinA-1)+1/(cosA+sinA+1)=cosecA+secA

If x cos A + sin A = m and
X sin A – y cos A = n, then prove that: x2 + y2 = m2 + n2

If m = a sec A + b tan A and n = a tan A + b sec A, then prove that : m2 - n2 = a2 - b2

If x = r sin A cos B, y = r sin A sin B and z = r cos A, then prove that: x2 + y2 + z2 = r2

If sin A + cos A = m and sec A + cosec A = n, show that: n (m2 - 1) = 2m

If x = r cos A cos B, y = r cos A sin B and Z = r sin A, show that:
x2 + y2 + z2 = r2

If cosA/cosB=m and cosA/sinB = n show that:

(m^2+n^2)cos^2B=n^2.

#### Chapter 21: Trigonometrical Identities Exercise 21C solutions [Page 0]

Solve.
cos22/sin68

Solve.
tan47/cot43

Solve.
sec75/(cosec15)

Solve.
cos55/sin35+cot35/tan55

solve.
cos240° + cos250°

solve.
sec18° - cot2 72°

Solve.
sin15° cos75° + cos15° sin75°

Solve.
sin42° sin48° - cos42° cos48°

Evaluate.
sin(90° - A) cosA + cos(90° - A) sinA

Evaluate.
sin235° + sin255°

Evaluate.
cot54^@/(tan36^@)+tan20^@/(cot70^@)-2

Evaluate.
(2tan53^@)/(cot37^@)-cot80^@/tan10^@

Evaluate.
cos225° + cos265° - tan245°

Evaluate.
(cos^2 32^@+cos^2 58^@)/(sin^2 59^@+sin^2 31^@)

Evaluate.
(sin77^@/cos13^@)^2+(cos77^@/sin13^@)-2cos^2 45^@

Evaluate.
cos^2 26^@+cos65^@sin26^@+tan36^@/cot54^@

Show that:
tan10° tan15° tan75° tan80° = 1

Show that:
sin 42° sec 48° + cos 42° cos ec48° = 2

Show that:
sin26^@/sec64^@+cos26^@/(cosec64^@)=1

Express the following in terms of angles between 0° and 45°:

sin59° + tan63°

Express the following in terms of angles between 0° and 45°:

cosec68° + cot72°

Express the following in terms of angles between 0° and 45°:

cos74° + sec67°

Show that:

sinA/sin(90^@-A)+cosA/cos(90^@-A)=secA cosecA

Show that:

sinAcosA-(sinAcos(90^@-A)cosA)/sec(90^@-A)-(cosAsin(90^@-A)sinA)/(cosec(90^@-A))=0

For triangle ABC, show that:

sin (A+B)/2=cosC/2

For triangle ABC, show that:

tan  (B+C)/2=cot  A/2

Evaluate:

3 sin72^@/(cos18^@)-sec32^@/(cosec58^@)

Evaluate:

3cos80° cosec10° + 2 cos59° cosec31°

Evaluate:

sin80^@/(cos10^@)+sin59^@ sec31^@

Evaluate:

cosec(65° + A) - sec(25° - A)

Evaluate:

2 tan57^@/(cot33^@)-cot70^@/(tan20^@)-sqrt2 cos45^@

Evaluate:

(cot^2 41^@)/(tan^2 49^@)-2 sin^2 75^@/cos^2 15^@

Evaluate:

cos70^@/(sin20^@)+cos59^@/(sin31^@)-8 sin^2 30^@

Evaluate:

14 sin30° + 6 cos60° - 5 tan45°

A triangle ABC is right angles at B; find the value of(secAcosecA-tanAcotC)/sinB

Find the value of x, if sin x = sin60° cos30° - cos60° sin30°

Find the value of x, if sin x = sin60° cos30° + cos60° sin30°

Find the value of x, if cos x = cos60° cos30° - sin60° sin30°

Find the value of x, if  tan x=(tan60^@-tan30^@)/(1+tan60^@tan30^@)

Find the value of x, if sin2x = 2sin 45° cos 45°

Find the value of x, if sin3x = 2sin 30° cos30°

Find the value of x, if cos(2x - 6) = cos230° - cos260°

find the value of angle A, where 0° ≤ A ≤ 90°.

sin(90° - 3A).cosec42° = 1

find the value of angle A, where 0° ≤ A ≤ 90°.

cos(90° - A).sec 77° = 1

Prove that:

(cos(90^@-theta)costheta)/cottheta=1-cos^2theta

Prove that:

(sinthetasin(90^@-theta))/cot(90^@-theta)=1-sin^2theta

Evaluate:

(sin35^@cos55^@+cos35^@sin55^@)/(cosec^2 10^@-tan^2 80^@)

#### Chapter 21: Trigonometrical Identities Exercise 21D solutions [Page 0]

Use tables to find sine of 21°

Use tables to find sine of 34° 42'

Use tables to find sine of 47° 32'

Use tables to find sine of 62° 57'

Use tables to find sine of 10° 20' + 20° 45'

Use tables to find cosine of 2° 4’

Use tables to find cosine of 8° 12’

Use tables to find cosine of 26° 32’

Use tables to find cosine of 65° 41’

Use tables to find cosine of 9° 23’ + 15° 54’

Use trigonometrical tables to find tangent of 37°

Use trigonometrical tables to find tangent of 42° 18'

Use trigonometrical tables to find tangent of 17° 27'

Use tables to find the acute angle θ, if the value of sin θ is 0.4848

Use tables to find the acute angle θ, if the value of sin θ is 0.3827

Use tables to find the acute angle θ, if the value of sin θ is 0.6525

Use tables to find the acute angle θ, if the value of cos θ is 0.9848

Use tables to find the acute angle θ, if the value of cos θ is 0.9574

Use tables to find the acute angle θ, if the value of cos θ is 0.6885

Use tables to find the acute angle θ, if the value of tan θ is 0.2419

Use tables to find the acute angle θ, if the value of tan θ is 0.4741

Use tables to find the acute angle θ, if the value of tan θ is 0.7391

#### Chapter 21: Trigonometrical Identities Exercise 21E, 21E 21E solutions [Page 0]

Prove the following identitie:

1/(cosA+sinA)+1/(cosA-sinA)=(2cosA)/(2cos^2A-1)

Prove the following identitie:

cosecA-cotA=sinA/(1+cosA

Prove the following identitie:

1-sin^2A/(1+cosA)=cosA

Prove the following identitie:

(1-cosA)/sinA+sinA/(1-cosA)=2 cosecA

Prove the following identitie:

cotA/(1-tanA)+tanA/(1-cotA)=1+tanA+cotA

Prove the following identitie:

cosA/(1+sinA)+tanA=secA

Prove the following identitie:

sinA/(1-cosA)-cotA=cosecA

Prove the following identitie:

(sinA-cosA+1)/(sinA+cosA-1)=cosA/(1-sinA)

Prove the following identitie:

sqrt((1+sinA)/(1-sinA))=cosA/(1-sinA)

Prove the following identitie:

sqrt((1-cosA)/(1+cosA))=sinA/(1+cosA)

Prove the following identitie:

(1+(secA-tanA)^2)/(cosecA(secA-tanA))=2tanA

Prove the following identitie:

((cosecA-cotA)^2+1)/(secA(cosecA-cotA))=2cotA

Prove the following identitie:

cot^2A((secA-1)/(1+sinA))+sec^2A((sinA-1)/(1+secA))=0

Prove the following identitie:

(1-2sin^2A)^2/(cos^4A-sin^4A)=2cos^2A-1

Prove the following identitie:

sec4A (1 - sin4A) - 2 tan2A = 1

Prove the following identitie:

cosec4A (1 - cos4A) - 2 cot2A = 1

Prove the following identitie:

(1 + tanA + secA)(1 + cotA - cosecA) = 2

If sinA + cosA = p

and secA + cosecA = q, then prove that: q(p2 - 1) = 2p

If x = a cosθ and y = b cotθ, show that:

a^2/x^2-b^2/y^2=1

If secA + tanA = p, show taht:

sinA = (p^2-1)/(p^2+1)

If tanA = n tanB and sinA = m sinB, prove that:

cos^2A=(m^2-1)/(n^2-1)

If 2 sin A – 1 = 0, show that:
Sin 3A = 3 sin A – 4 sin3A

If 4 cos2 A – 3 = 0, Show that:
cos 3 A = 4 cos3 A – 3 cos A

Evaluate

2(tan35^@/cot55^@)+(cot55^@/tan35^@)-3(sec40^@/(cosec50^@))

Evaluate

sec26^@ sin64^@+(cosec33^@)/sec57^@

Evaluate

(5sin66^@)/(cos24^@)-(2cot85^@)/tan5^@

Evaluate

cos40° cosec50° + sin50° sec40°

Evaluate

sin27° sin63° - cos63° cos27°

Evaluate

(3sin72^@)/(cos18^@)-sec32^@/(cosec58^@)

Evaluate

3 cos80° cosec10°+ 2 cos59° cosec31°

Evaluate

cos75^@/(sin15^@)+sin12^@/(cos78^@)-cos18^@/sin72^@

Prove that:

tan(55° + x) = cot(35° - x)

Prove that:

sec(70° - θ) = cosec(20° + θ)

Prove that:

sin(28° + A) = cos(62° - A)

Prove that:

1/(1+cos(90^@ - A))+ 1/(1-cos(90^@-A))=2cosec^2(90^@-A)

Prove that:

1/(1+sin(90^@-A))+1/(1-sin(90^@-A))=2sec^2(90^@-A)

If A and B are complementary angles, prove that:

cotB + cosB = secA cosB (1 + sinB)

If A and B are complementary angles, prove that:

cotA cotB - sinA cosB  - cosA sinB = 0

If A and B are complementary angles, prove that:

cosec2A + cosec2B = cosec2A cosec2B

If A and B are complementary angles, prove that:

(sinA+sinB)/(sinA-sinB)+(cosB-cosA)/(cosB+cosA)=2/(2sin^2A-1)

Prove that

1/(sinA-cosA)-1/(sinA+cosA)=(2cosA)/(2sin^2A-1)

Prove that

cot^2A/(cosecA-1)-1=cosecA

Prove that

cosA/(1+sinA)=secA-tanA

Prove that

cosA (1 + cotA) + sinA (1 + tanA = secA + cosecA

Prove that

(sinA-cosA)(1+tanA+cotA)=secA/(cosec^2A)-(cosecA)/sec^2A

Prove that

sqrt(sec^2A+cosec^2A)=tanA + cotA

Prove that

(sinA + cosA) (secA + cosecA) = 2 + secA cosecA

Prove that

(tanA + cotA) (cosecA - sinA) (secA - cosA) = 1

Prove that

cot^2A-cot^2B=(cos^2A-cos^2B)/(sin^2Asin^2B)=cosec^2A-cosec^2B

If 4 cos2 A – 3 = 0 and ≤ A ≤ 90°, then prove that :
sin 3 A = 3 sin A – 4 sin3 A

If 4 cos2 A – 3 = 0 and ≤ A ≤ 90°, then prove that:
cos 3 A = 4 cos3 A – 3 cos A

Find A, if 0° ≤ A ≤ 90° and 2cos2A - 1 = 0

Find A, if 0° ≤ A ≤ 90° and sin 3A - 1 = 0

Find A, if 0° ≤ A ≤ 90° and 4sin2A - 3 = 0

Find A, if 0° ≤ A ≤ 90° and cos2A - cosA = 0

Find A, if 0° ≤ A ≤ 90° and 2cos2A + cosA - 1 = 0

If 0° < A < 90°; Find A, if :

(cos A )/(1 - sin A) + (cos A)/(1 + sin A) = 4

If 0° < A < 90°; Find A, if sinA/(secA-1)+sinA/(secA+1)=2

Prove that:

(cosecA - sinA) (secA - cosA) sec2A = tanA

## Chapter 21: Trigonometrical Identities

21A21B21C21D21E21E 21E

#### Selina Selina ICSE Concise Mathematics Class 10 (2018-2019) ## Selina solutions for Class 10 Mathematics chapter 21 - Trigonometrical Identities

Selina solutions for Class 10 Maths chapter 21 (Trigonometrical Identities) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CISCE Selina ICSE Concise Mathematics for Class 10 (2018-2019) solutions in a manner that help students grasp basic concepts better and faster.

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Concepts covered in Class 10 Mathematics chapter 21 Trigonometrical Identities are Trigonometric Ratios of Complementary Angles, Trigonometric Identities, Heights and Distances - Solving 2-D Problems Involving Angles of Elevation and Depression Using Trigonometric Tables, Trigonometry Problems and Solutions.

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