Q. 15 H5.0( 3 Votes )

# Prove the following identities :sin4 A – cos4 A = 2 sin2 A – 1 = 1 – 2 cos2 A = sin2 A – cos2 A

Answer :

Given:

Taking I term

= sin4 A – cos4 A I term

= (sin2 A)2 – (cos2 A)2

= (sin2 A – cos2 A)(sin2 A+ cos2 A )

[ (a2 – b2) = (a + b) (a – b)]

= (sin2 A – cos2 A)(1) [ cos2 θ + sin2 θ = 1]

= (sin2 A – cos2 A) …(i) IV term

From Eq. (i)

= {sin2 A – (1 – sin2 A)} [ cos2 θ + sin2 θ = 1]

= sin2 A – 1 + sin2 A

= 2 sin2 A – 1 II term

Again, From Eq. (i)

= {(1 – cos2 A) – cos2 A } [ cos2 θ + sin2 θ = 1]

=1 – 2 cos2 A III term

Hence, I = II = III = IV

Hence Proved

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