Q. 15 J4.8( 4 Votes )

Prove the following identities :

2 cos2 θ – cos4 θ + sin4 θ = 1

Answer :

Taking LHS = 2 cos2 θ – cos4 θ + sin4 θ

= 2 cos2 θ – (cos4 θ – sin4 θ)


= 2 cos2 θ – [(cos2 θ)2 – (sin2 θ)2]


Using identity, (a2 – b2) = (a + b) (a – b)


= 2 cos2 θ – [(cos2 θ – sin2 θ)(cos2 θ+ sin2 θ )]


= 2 cos2 θ – [(cos2 θ – sin2 θ)(1)] [ cos2 θ + sin2 θ = 1]


=2 cos2 θ – cos2 θ + sin2 θ


= cos2 θ + sin2 θ


= 1 [ cos2 θ + sin2 θ = 1]


= RHS


Hence Proved


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