Question

$\sf{ {sin}^{4} \: \theta \: - {cos}^{4} \: \theta = 1 - 2 \: {cos}^{2} \: \theta}$

Prove it ​

Answer 1

How to prove cos^4 x - sin^4 x + 1 = 2 cos^2 x?

LHS = cos^4 x - sin^4 x + 1

= (cos^2 x + sin^2 x)(cos^2 x - sin^2 x) + 1

= cos^2 x - sin^2 x + 1 [Because cos^2 x + sin^2 x = 1]

= 2 cos^2 x [Because 1 - sin^2 x = cos^2 x ]

= RHS

Proved.

Answer 2

Observe that

sin4θ+cos4θ=1−2sin2θcos2θ⟺sin4θ+cos4θ+2sin2θcos2θ=1

sin4⁡θ+cos4⁡θ=1−2sin2⁡θcos2⁡θ⟺sin4⁡θ+cos4⁡θ+2sin2⁡θcos2⁡θ=1

which is always true since

sin4θ+cos4θ+2sin2θcos2θ=(sin2θ+cos2θ)2=12=1

sin4⁡θ+cos4⁡θ+2sin2⁡θcos2⁡θ=(sin2⁡θ+cos2⁡θ)2=12=1

so the identity is proved.