Selina solutions for Mathematics [English] Class 10 ICSE chapter 21 - Trigonometrical Identities [Latest edition]

Prove the following identities:

`(sec A - 1)/(sec A + 1) = (1 - cos A)/(1 + cos A)`

Prove the following identities:

`(1 + sin A)/(1 - sin A) = (cosec  A + 1)/(cosec  A - 1)`

Prove the following identities:

`1/(tan A + cot A) = cos A sin A`

Prove the following identities:

`tan A - cot A = (1 - 2cos^2A)/(sin A cos A)`

Prove the following identities:

sin⁴A – cos⁴A = 2sin²A – 1

Prove the following identities:

(1 – tan A)² + (1 + tan A)² = 2 sec²A

Prove the following identities:

cosec⁴ A – cosec² A = cot⁴ A + cot² A

Prove the following trigonometric identities.

sec A (1 − sin A) (sec A + tan A) = 1

Prove the following identities:

cosec A(1 + cos A) (cosec A – cot A) = 1

Prove the following identities:

sec² A + cosec² A = sec² A . cosec² A

Prove the following identities:

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

Prove the following identities:

tan² A – sin² A = tan² A . sin² A

Prove the following identities:

cot² A – cos² A = cos² A . cot² A

Prove the following identities:

(cosec A + sin A) (cosec A – sin A) = cot² A + cos² A

Prove the following identities:

(sec A – cos A) (sec A + cos A) = sin² A + tan² A 

Prove the following identities:

(cos A + sin A)² + (cos A – sin A)² = 2

Prove the following identities:

(cosec A – sin A) (sec A – cos A) (tan A + cot A) = 1

Prove the following identities:

`1/(secA + tanA) = secA - tanA`

Prove the following identities:

`cosecA + cotA = 1/(cosecA - cotA)`

Prove the following identities:

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

Prove the following identities:

(sin A + cosec A)² + (cos A + sec A)² = 7 + tan² A + cot² A

Prove the following identities:

sec² A . cosec² A = tan² A + cot² A + 2

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

`(1 + sinA)/cosA + cosA/(1 + sinA) = 2secA`

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

`(sintheta - 2sin^3theta)/(2cos^3theta - costheta) = tantheta`

Prove the following identities:

`sinA/(1 + cosA) = cosec A - cot A`

Prove the following identities:

`cosA/(1 - sinA) = sec A + tan A`

Prove the following identities:

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

Prove the following identities:

(1 + cot A – cosec A)(1 + tan A + sec A) = 2

Prove the following identities:

`sqrt((1 + sinA)/(1 - sinA)) = sec A + tan A`

Prove the following identities:

`sqrt((1 - cosA)/(1 + cosA)) = cosec A - cot A`

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

`(1+ sin A)/(cosec A - cot A) - (1 - sin A)/(cosec A + cot A) = 2(1 + cot A)`

Prove the following identities:

`(costhetacottheta)/(1 + sintheta) = cosectheta - 1`

Prove that:

(sec A − tan A)² (1 + sin A) = (1 − sin A)

Prove that:

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

Prove that:

`tanA/(1 - cotA) + cotA/(1 - tanA) = secA cosecA + 1`

Prove that:

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

Prove that:

2 sin² A + cos⁴ A = 1 + sin⁴ A 

Prove that:

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

Prove that:

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

Prove that:

(1 + tan A . tan B)² + (tan A – tan B)² = sec² A sec² B

Prove that:

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

If x cos A + y 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) = 2 m

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 : (m² + n²) cos² B = n².

Show that : tan 10° tan 15° tan 75° tan 80° = 1

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

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

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

sin 59° + tan 63°

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^circ - A) + cosA/cos(90^circ - A) = sec A cosec A`

Show that : `sinAcosA - (sinAcos(90^circ - A)cosA)/sec(90^circ - A) - (cosAsin(90^circ - A)sinA)/(cosec(90^circ - A)) = 0`

For triangle ABC, show that : `sin  (A + B)/2 = cos  C/2`

For triangle ABC, show that : `tan  (B + C)/2 = cot  A/2`

Evaluate:

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

Evaluate:

3cos80° cosec10° + 2 sin59° sec31°

Evaluate:

`sin80^circ/(cos10^circ) + sin59^circ  sec31^circ`

Prove that:

tan (55° - A) - cot (35° + A)

Evaluate:

cosec (65° + A) – sec (25° – A)

Evaluate:

`2 tan57^circ/(cot33^circ) - cot70^circ/(tan20^circ) - sqrt(2)  cos45^circ`

Evaluate:

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

Evaluate:

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

Evaluate:

14 sin 30° + 6 cos 60° – 5 tan 45°

A triangle ABC is right angles at B; find the value of`(secA.cosecC - tanA.cotC)/sinB`

Find the value of x, if sin x = sin 60° cos 30° – cos 60° sin 30°

Find the value of x, if sin x = sin 60° cos 30° + cos 60° sin 30°

Find the value of x, if cos x = cos 60° cos 30° – sin 60° sin 30°

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

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

Find the value of x, if sin 3x = 2 sin 30° cos 30°

Find the value of x, if cos (2x – 6) = cos² 30° – cos² 60°

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

sin (90° – 3A) . cosec 42° = 1

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

cos (90° – A) . sec 77° = 1

Prove that:

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

Prove that:

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

Evaluate:

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

Evaluate:

sin² 34° + sin² 56° + 2 tan 18° tan 72° – cot² 30°

Without using trigonometrical tables, evaluate:

`cosec^2 57^circ - tan^2 33^circ + cos 44^circ cosec 46^circ - sqrt(2) cos 45^circ - tan^2 60^circ`

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

`cosA/(1 + sinA) + tanA = secA`

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

sec⁴ A (1 – sin⁴ A) – 2 tan² A = 1

Prove the following identities:

cosec⁴ A (1 – cos⁴ A) – 2 cot² A = 1

Prove the following identities:

(1 + tan A + sec A) (1 + cot A – cosec A) = 2

If sin A + cos A = p and sec A + cosec A = q, then prove that : q(p² – 1) = 2p.

If x = a cos θ and y = b cot θ, show that:

`a^2/x^2 - b^2/y^2 = 1` 

If sec A + tan A = p, show that:

`sin A = (p^2 - 1)/(p^2 + 1)`

If tan A = n tan B and sin A = m sin B, prove that:

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

If 2 sin A – 1 = 0, show that: sin 3A = 3 sin A – 4 sin³ A

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

Evaluate:

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

Evaluate:

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

Evaluate:

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

Evaluate:

cos 40° cosec 50° + sin 50° sec 40°

Evaluate:

sin 27° sin 63° – cos 63° cos 27°

Evaluate:

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

Evaluate:

3 cos 80° cosec 10° + 2 cos 59° cosec 31°

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:

cot B + cos B = sec A cos B (1 + sin B)

If A and B are complementary angles, prove that:

cot A cot B – sin A cos B – cos A sin B = 0

If A and B are complementary angles, prove that:

cosec² A + cosec² B = cosec² A cosec² B

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:

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

Prove that:

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

Prove that:

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

Prove that:

(sin A + cos A) (sec A + cosec A) = 2 + sec A cosec A

Prove that:

(tan A + cot A) (cosec A – sin A) (sec A – cos A) = 1

Prove that

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

Prove that:

`(cot A - 1)/(2 - sec^2 A) = cot A/(1 + tan A)` 

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

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

Find A, if 0° ≤ A ≤ 90° and 2 cos² A – 1 = 0

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

Find A, if 0° ≤ A ≤ 90° and 4 sin² A – 3 = 0

Find A, if 0° ≤ A ≤ 90° and cos² A – cos A = 0

Find A, if 0° ≤ A ≤ 90° and 2 cos² A + cos A – 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:

(cosec A – sin A) (sec A – cos A) sec² A = tan A

Prove the identity (sin θ + cos θ)(tan θ + cot θ) = sec θ + cosec θ.

Evaluate without using trigonometric tables, 

`sin^2 28^@ + sin^2 62^@ + tan^2 38^@ - cot^2 52^@ + 1/4 sec^2 30^@`