Is cos^4(β)-sin^4(β)=2cos^2(β)-1 ?

The answer to whether cos^4(β)-sin^4(β)=2cos^2(β)-1 is True

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prove cos^4(β)-sin^4(β)=2cos^2(β)-1

prove

cos^{4} (β) - sin^{4} (β) = 2 cos^{2} (β) - 1

True

Solution steps

cos^{4} (β) - sin^{4} (β) = 2 cos^{2} (β) - 1

Manipulating left side

cos^{4} (β) - sin^{4} (β)

Factor

cos^{4} (β) - sin^{4} (β) : (cos^{2} (β) + sin^{2} (β)) (cos (β) + sin (β)) (cos (β) - sin (β))

= (cos (β) + sin (β)) (cos (β) - sin (β)) (cos^{2} (β) + sin^{2} (β))

Rewrite using trig identities

= 2 cos^{2} (β) - 1

We showed that the two sides could take the same form

\Rightarrow True

(1 - sin^{2} (x)) \cdot csc^{2} (x) = cot^{2} (x)

sec^{2} (x) (1 - cos^{2} (x)) = tan^{2} (x)

cot^{2} (θ) - cos^{2} (θ) = cot^{2} (θ) cos^{2} (θ)

\frac{sin ^{2} ( - x )}{tan ^{2} ( x )} = cos^{2} (x)

\frac{1 + cos ( θ )}{cos ( θ )} = sec (θ) + 1

Is cos^4(β)-sin^4(β)=2cos^2(β)-1 ?
The answer to whether cos^4(β)-sin^4(β)=2cos^2(β)-1 is True