Q15 of 268 Page 5

Prove the following identities :

cos4 θ – sin4 θ = cos2 θ – sin2 θ = 2 cos2 θ – 1

Given:


Taking I term


= cos4 θ – sin4 θ → I term


= (cos2 θ)2 – (sin2 θ)2


= (cos2 θ – sin2 θ)(cos2 θ+ sin2 θ )


[∵ (a2 – b2) = (a + b) (a – b)]


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


= (cos2 θ – sin2 θ) …(i) → II term


From Eq. (i)


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


= 2 cos2 θ – 1 → III term


Hence, I = II = III


Hence Proved


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