Question

show that sin⁴ theta + Cos⁴ theta =1- 2 sin² theta cos² theta​

Answer 1

Step-by-step explanation:

let θ=α, then we have,

$\sin^{4} ( \alpha ) + \cos^{4} ( \alpha ) = {( \sin ^{2} ( \alpha ) + \cos ^{2} ( \alpha ) )}^{2} - 2 \sin^{2} ( \alpha ) \cos^{2} ( \alpha )$

as: we know that (a²+b²)²=a^4 + b^4 +2a²b²

so,

$\sin^{4} ( \alpha ) + \cos^{4} ( \alpha ) = {(1)}^{2} - 2 \sin^{2} ( \alpha ) \cos^{2} ( \alpha )$

as: sin²(θ) + cos²(θ)= 1

$= > \sin^{4} ( \alpha ) + \cos^{4} ( \alpha ) = 1 - 2 \sin^{2} ( \alpha ) \cos^{2} ( \alpha )$

on replacing α with θ, we will obtain the above result