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# Prove the following identities - CBSE Class 10 - Mathematics

#### Question

Prove the following identities:

(i) cos4^4 A – cos^2 A = sin^4 A – sin^2 A

(ii) cot^4 A – 1 = cosec^4 A – 2cosec^2 A

(iii) sin^6 A + cos^6 A = 1 – 3sin^2 A cos^2 A.

#### Solution

(i) We have,

LHS = cos4^4 A – cos^2 A = cos^2 A (cos^2 A – 1)

= – cos^2 A (1 – cos^2 A) = – cos^2 A sin2A

= –(1 – sin^2 A) sin^2 A = – sin^2 A + sin^4 A

= sin^4 A – sin^2 A = RHS

(ii) We have,

LHS = cot^4 A – 1 = (cosec^2 A – 1)^2 – 1

[∵ cot^2 A = cosec^2 A –1 ⇒ cot^4 A = (cosec^2 A – 1)^2 ]

= cosec^4 A – 2 cosec^2 A + 1 – 1

= cosec^4 A – 2cosec^2 A = RHS

(iii) We have,

LHS = sin^6 A + cos^6 A = (sin^2 A)^3 + (cos2 A)^3

= (sin^2 A + cos^2 A) {(sin^2 A)^2 + (cos^2 A)^2 – sin^2 A cos^2 A)}

[∵ a^3 + b^3 = (a + b) (a^2 – ab + b^2 )]

={(sin^2 A)^2 + (cos^2 A)^2 + 2 sin^2 A cos^2 A – sin^2 A cos^2 A}

= [(sin^2 A + cos^2 A)^2 – 3 sin^2 A cos^2 A]

= 1 – 3sin^2 A cos^2 A = RHS

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Solution Prove the following identities Concept: Trigonometric Identities.
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