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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