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# Frank solutions for Class 10 Maths chapter 21 - Trigonometric Identities

## Chapter 21: Trigonometric Identities

Exercise 21.1Exercise 21.2Exercise 21.3

#### Frank solutions for Class 10 Maths Chapter 21 Exercise Exercise 21.1 [Page 0]

Prove the following identity :

(1 - sin^2θ)sec^2θ = 1

Prove the following identity :

(1 - cos^2θ)sec^2θ = tan^2θ

Prove the following identity :

tanA+cotA=secAcosecA

Prove the following identity :

sinθ(1 + tanθ) + cosθ(1 +cotθ) = secθ + cosecθ

Prove the following identity :

( 1 + cotθ - cosecθ) ( 1 + tanθ + secθ)

Prove the following identity :

sinθcotθ + sinθcosecθ = 1 + cosθ

Prove the following identity :

secA(1 - sinA)(secA + tanA) = 1

Prove the following identity :

secA(1 + sinA)(secA - tanA) = 1

Prove the following identity :

cosecθ(1 + cosθ)(cosecθ - cotθ) = 1

Prove the following identity :

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

Prove the following identity :

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

Prove the following identity :

cosA/(1 + sinA) = secA - tanA

Prove the following identity :

(tanθ + secθ - 1)/(tanθ - secθ + 1) = (1 + sinθ)/(cosθ)

Prove the following identity :

sin^2Acos^2B - cos^2Asin^2B = sin^2A - sin^2B

Prove the following identity :

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

Prove the following identity :

cosec^4A - cosec^2A = cot^4A + cot^2A

Prove the following identity :

sec^2A + cosec^2A = sec^2Acosec^2A

Prove the following identity :

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

Prove the following identity :

tan^2A - sin^2A = tan^2A.sin^2A

Prove the following identity :

(secA - cosA)(secA + cosA) = sin^2A + tan^2A

Prove the following identity :

(cosA + sinA)^2 + (cosA - sinA)^2 = 2

Prove the following identity :

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

Prove the following identity :

sec^2A.cosec^2A = tan^2A + cot^2A + 2

Prove the following identity :

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

Prove the following identity :

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

Prove the following identity :

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

Prove the following identity :

sin^4A + cos^4A = 1 - 2sin^2Acos^2A

Prove the following Identities :

(cosecA)/(cotA+tanA)=cosA

Prove the following identities:

(tan"A"+tan"B")/(cot"A"+cot"B")=tan"A"tan"B"

Prove the following identities:

(sec"A"-1)/(sec"A"+1)=(sin"A"/(1+cos"A"))^2

Prove the following identity :

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

Prove the following identity :

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

Prove the following identity :

(cotA + tanB)/(cotB + tanA) = cotAtanB

Prove the following identity :

1/(tanA + cotA) = sinAcosA

Prove the following identity :

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

Prove the following identity :

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

Prove the following identity :

cosecA + cotA = 1/(cosecA - cotA)

Prove the following identity :

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

Prove the following identity :

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

Prove the following identity :

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

Prove the following identity :

sqrt(cosec^2q - 1) = "cosq  cosecq"

Prove the following identity :

sqrt((1 + sinq)/(1 - sinq)) + sqrt((1- sinq)/(1 + sinq)) = 2secq

Prove the following identity :

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

Prove the following identity :

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

Prove the following identity :

sqrt((secq - 1)/(secq + 1)) + sqrt((secq + 1)/(secq - 1)) = 2 cosesq

Prove the following identity :

(secθ - tanθ)^2 = (1 - sinθ)/(1 + sinθ)

Prove the following identity :

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

Prove the following identity :

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

Prove the following identity :

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

Prove the following identity :

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

Prove the following identity :

(1 + tan^2A) + (1 + 1/tan^2A) = 1/(sin^2A - sin^4A)

Prove the following identity :

(cos^3A + sin^3A)/(cosA + sinA) + (cos^3A - sin^3A)/(cosA - sinA) = 2

Prove the following identity :

(tanθ + 1/cosθ)^2 + (tanθ - 1/cosθ)^2 = 2((1 + sin^2θ)/(1 - sin^2θ))

Prove the following identity :

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

Prove the following identity :

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

Prove the following identity :

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

Prove the following identity :

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

Prove the following identity  :

(1 + cotA)^2 + (1 - cotA)^2 = 2cosec^2A

Prove the following identity :

(cosecθ)/(tanθ + cotθ) = cosθ

Prove the following identity :

(1 + tan^2θ)sinθcosθ = tanθ

Prove the following identity :

(1 + sinθ)/(cosecθ - cotθ) - (1 - sinθ)/(cosecθ + cotθ) = 2(1 + cotθ)

Prove the following identity :

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

Prove the following identity :

2(sin^6θ + cos^6θ) - 3(sin^4θ + cos^4θ) + 1 = 0

Prove the following identity :

sin^8θ - cos^8θ = (sin^2θ - cos^2θ)(1 - 2sin^2θcos^2θ)

Prove the following identity :

sec^4A - sec^2A = sin^2A/cos^4A

Prove the following identity :

tan^2θ/(tan^2θ - 1) + (cosec^2θ)/(sec^2θ - cosec^2θ) = 1/(sin^2θ - cos^2θ)

Prove the following identity :

(sec^2θ - sin^2θ)/tan^2θ = cosec^2θ - cos^2θ

Prove the following identity :

(cos^3θ + sin^3θ)/(cosθ + sinθ) + (cos^3θ - sin^3θ)/(cosθ - sinθ) = 2

Prove the following identity :

(tanθ + sinθ)/(tanθ - sinθ) = (secθ + 1)/(secθ - 1)

Prove the following identity :

[1/((sec^2θ - cos^2θ)) + 1/((cosec^2θ - sin^2θ))](sin^2θcos^2θ) = (1 - sin^2θcos^2θ)/(2 + sin^2θcos^2θ)

Prove the following identity :

(cot^2θ(secθ - 1))/((1 + sinθ)) = sec^2θ((1-sinθ)/(1 + secθ))

#### Frank solutions for Class 10 Maths Chapter 21 Exercise Exercise 21.2 [Page 0]

If m = a secA + b tanA and n = a tanA + b secA , prove that m2 - n2 = a2 - b2

If x/(a cosθ) = y/(b sinθ)   "and"  (ax)/cosθ - (by)/sinθ = a^2 - b^2 , "prove that"  x^2/a^2 + y^2/b^2 = 1

If x = r sinA cosB , y = r sinA sinB and z = r cosA , prove that   x^2 + y^2 + z^2 = r^2

If sinA + cosA = m and secA + cosecA = n , prove that n(m2 - 1) = 2m

If x = acosθ , y = bcotθ , prove that a^2/x^2 - b^2/y^2 = 1.

If secθ + tanθ = m , secθ - tanθ = n , prove that mn = 1

If x = asecθ + btanθ and y = atanθ + bsecθ , prove that x^2 - y^2 = a^2 - b^2

If tanA + sinA = m and tanA - sinA = n , prove that (m^2 - n^2)^2 = 16mn

If sinA + cosA = sqrt(2) , prove that sinAcosA = 1/2

If asin^2θ + bcos^2θ = c and p sin^2θ + qcos^2θ = r , prove that (b - c)(r - p) = (c - a)(q - r)

#### Frank solutions for Class 10 Maths Chapter 21 Exercise Exercise 21.3 [Page 0]

Without using trigonometric table , evaluate :

cosec49°cos41° + (tan31°)/(cot59°)

Without using trigonometric table , evaluate :

(sin47^circ/cos43^circ)^2 - 4cos^2 45^circ + (cos43^circ/sin47^circ)^2

Without using trigonometric table , evaluate :

cos90^circ + sin30^circ tan45^circ cos^2 45^circ

Without using trigonometric table , evaluate :

(sin49^circ/sin41^circ)^2 + (cos41^circ/sin49^circ)^2

Without using trigonometric table , evaluate :

sin72^circ/cos18^circ  - sec32^circ/(cosec58^circ)

Find the value of θ(0^circ < θ < 90^circ) if :

cos 63^circ sec(90^circ - θ) = 1

Find the value of θ(0^circ < θ < 90^circ) if :

tan35^circ cot(90^circ - θ) = 1

Without using trigonometric identity , show that :

sin42^circ sec48^circ + cos42^circ cosec48^circ = 2

Without using trigonometric identity , show that :

tan10^circ tan20^circ tan30^circ tan70^circ tan80^circ = 1/sqrt(3)

Without using trigonometric identity , show that :

sin(50^circ + θ) - cos(40^circ - θ) = 0

Without using trigonometric identity , show that :

cos^2 25^circ + cos^2 65^circ = 1

Without using trigonometric identity , show that :

sec70^circ sin20^circ - cos20^circ cosec70^circ = 0

Prove that sinA/sin(90^circ - A) + cosA/cos(90^circ - A) = sec(90^circ - A) cosec(90^circ - A)

For ΔABC , prove that :

tan ((B + C)/2) = cot "A/2

For ΔABC , prove that :

sin((A + B)/2) = cos"C/2

Prove that  sin(90^circ - A).cos(90^circ - A) = tanA/(1 + tan^2A)

Find the value of x , if cosx = cos60^circ cos30^circ - sin60^circ sin30^circ

Find x , if cos(2x - 6) = cos^2 30^circ - cos^2 60^circ

Given cos38^circ sec(90^circ - 2A) = 1 , Find the value of <A

prove that 1/(1 + cos(90^circ - A)) + 1/(1 - cos(90^circ - A)) = 2cosec^2(90^circ - A)

## Chapter 21: Trigonometric Identities

Exercise 21.1Exercise 21.2Exercise 21.3

## Frank solutions for Class 10 Mathematics chapter 21 - Trigonometric Identities

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Concepts covered in Class 10 Mathematics chapter 21 Trigonometric 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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