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

sin⁴ θ − cos⁴ θ is equal to ______.

(1 + tan θ)² + (1 − tan θ)² is equal to ______.

cot⁴ θ + cot² θ is equal to ______.

\[\frac{1}{1 - \sin A}\] is equal to ______.

\[\frac{\tan^2 A}{(\sec A + 1)^2}\] is equal to ______.

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:

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:

sec²A + cosec²A = sec²A . cosec²A

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:

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

Prove the following identities:

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

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:

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

Prove the following identities:

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

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:

`(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:

`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:

`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:

`(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`

(1 + cot² A) + (1 + tan² A) is equal to ______.

If a = tan θ and b = sec θ, the relation between a and b is ______.

\[\frac{1}{\cot \theta + \tan \theta}\] is equal to ______.

(sec θ − cos θ)² − (sec θ + cos θ)² is equal to ______.

(cot A − cot B)² + (1 + cot A cot B)² is equal to ______.

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  "cosec"  A + 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 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.

sin² A + sin²(90° − A) is equal to ______.

In a triangle ABC, \[ \sec \frac{C + A}{2}\] is equal to ______.

\[\frac{\cot 46^\circ}{\tan 44^\circ} - 3 \frac{\sec 20^\circ}{\operatorname{cosec} 70^\circ} + 5\] is equal to ______.

sin 67° . cos 23° + cos 67° . sin 23° is equal to ______.

\[\frac{\cos \theta \cdot \cos(90^\circ - \theta)}{\tan(90^\circ - \theta)}\] is equivalent to ______.

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

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

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

sin 59° + tan 63°

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

cosec 68° + cot 72°

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`

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

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`

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`

If \[2\sin \theta = \sqrt{3}\]; then:

If cos 62° 40' = x, the value of 1 + sin² 27° 20' is ______.

sin θ = 0.5 implies tan θ + cot θ is equal to:

If cos 63° 36' = 0.4446, the value of sin 26° 24' is ______.

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 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 trigonometrical tables to find tangent of 37°

Use trigonometrical tables to find tangent of 42° 18'

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 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 tan θ is 0.2419

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

For the given figure, which of the following conditions is true:

If \[\sin A = \frac{4}{5}\], the value of \[\sqrt{\frac{1 - \cos(90^\circ - A)}{1 + \cos(90^\circ - A)}}\] is ______.

(1 + tan² A) × cos² A − 1 is ______.

\[\frac{\sin A \tan A}{1 - \cos^2 A}\] is equal to:

For acute angles α and β, if α > β, then:

A: cos α > cos β

B: tan α > tan β

C: sin α > sin β

D: cot α > cot β

Which of the above statement/statements is/are true:

If M = x cos A + y sin A and N = x sin A − y cos A, then M² + N² is ______.

\[\frac{\tan 60^\circ - \tan 30^\circ}{1 + \tan 60^\circ \tan 30^\circ}\] is equal to ______.

\[3 \frac{\sin 47^\circ}{\cos 43^\circ} - \frac{\tan 50^\circ}{\cot 40^\circ} - 2\sin^2 45^\circ\] is equal to:

If sin 2x = 2 sin 45° cos 45°; the value of x is ______.

For acute angle θ; sec² θ = 1 − tan² θ.

Assertion (A): sec² θ = 1 − tan² θ is not a trigonometric identity.

Reason (R): For an acute angle θ, the trigonometric equation is an identity, if it is satisfied for every value of angle θ.

(1+ tan² A) (1 – sin A) (1 + sin A)

Assertion (A): The value of given trigonometric expression is [0.]

Reason (R): The given expression is equal to sec² A . cos² A.

\[\cos A = \frac{\sqrt{3}}{2}\] and \[\sin B = \frac{1}{2}\]

Assertion (A): \[\tan(A + B) = \sqrt{3}\]

Reason (R): \[\cos A = \frac{\sqrt{3}}{2} \Rightarrow A = 30^\circ\]

\[\sin B = \frac{1}{2} \Rightarrow B = 30^\circ\]

x = (cosec A + cot A) (1 − cos A).

Assertion (A): x = sin A.

Reason (R): \[x = \left(\frac{1}{\sin A} + \frac{\cos A}{\sin A}\right)(1 - \cos A)\]

\[{} = \frac{\sin^2 A}{\sin A} = \sin A\]

\[\frac{1 - \sin A}{\cos A} = \sec A - \tan A.\]

Statement (1): \[\frac{1 - \sin A}{\cos A} = \sec A - \tan A\]

\[\Rightarrow \frac{1 + \sin A}{\cos A} = \sec A + \tan A\]

Statement (2): \[\frac{1 - \sin A}{\cos A} - \frac{1 + \sin A}{\cos A} = 2\sec A\]

\[\cos^2 \theta + \frac{1}{1 + \cot^2 \theta} = x.\]

Statement (1): x = 1.

Statement (2): \[x = \cos^2 \theta + \frac{1}{\operatorname{cosec}^2 \theta}\]

\[{} = \cos^2 \theta + \sin^2 \theta\]

\[\frac{\sin^2 \theta}{\cos^2 \theta} - \frac{1}{\cos^2 \theta} = x.\]

Statement (1): x = 1.

Statement (2): \[x = \frac{\sin^2 \theta - 1}{\cos^2 \theta} = \frac{-\cos^2 \theta}{\cos^2 \theta} = -1\]

sin A = cos A and tan² A + cot² A + 2.

Statement (1): A = 45°

tan² A + cot² A + 2 = 4

Statement (2): sin A = cos A ⇒ A = 45°

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

Prove the following identities:

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

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:

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

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)`

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)`

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:

`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^@`

Prove that:

(1+ cot² A) (1 − cos A) (1 + cos A) = 1

Prove that:

\[\frac{\sin^4 A + \cos^4 A}{\sin^2 A \cdot \cos^2 A} = \tan^2 A + \cot^2 A\]

Prove that:

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

Prove that:

\[\frac{\cos^2 A}{1 - \tan A} + \frac{\sin^3 A}{\sin A - \cos A} = 1 + \sin A \cos A\]

Prove that:

\[\frac{\sin A + 1 - \cos A}{\cos A - 1 + \sin A} = \frac{1 + \sin A}{\cos A}\]

If sin A + cos A = \[\sqrt3\]; prove that: tan A + cot A = 1

Prove that:

\[\frac{\sin A}{\sin (90^\circ - A)} + \frac{\cos A}{\cos (90^\circ - A)} = \sec A \operatorname{cosec} A\]

Prove that:

\[2 \cos^2 A + \frac{2}{1 + \cot^2 A} = 2\]

Evaluate:

sec² A(1 + sin A) (1 – sin A)

Evaluate:

\[x^2 - \frac{1}{x^2}\] when \[\operatorname{cosec} A = 2x\] and \[\tan A = \frac{x}{2}\]

In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B produce the same (equal) value for all values of the variable within a certain domain.

Infact, identity in mathematics is an equation that is true for all values of its variable. 

For angle x = 30°:

\[A = \sin^2 x = \sin^2 30^\circ = \left(\frac{1}{2}\right)^2 = \frac{1}{4}\]

\[B = 1 - \cos^2 x = 1 - \cos^2 30^\circ\]

\[{} = 1 - \left(\frac{\sqrt{3}}{2}\right)^2 = 1 - \frac{3}{4} = \frac{1}{4}\]

That is, A = B

⇒ sin² x = 1 − cos² x is an identity

Similarly, for angle x = 60°: 

1. Find the value of tan x + cot x (say, A)
2. Find the value of \[\frac{1}{\sin x \cos x}\] (say B)
3. Write the relation between A and B.
4. Write the corresponding identity.