Is cos^4(A)-sin^4(A)=cos(2A) ?

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

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prove cos^4(A)-sin^4(A)=cos(2A)

prove

cos^{4} (A) - sin^{4} (A) = cos (2 A)

True

Solution steps

cos^{4} (A) - sin^{4} (A) = cos (2 A)

Manipulating left side

cos^{4} (A) - sin^{4} (A)

Factor

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

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

Rewrite using trig identities

= cos (2 A)

We showed that the two sides could take the same form

\Rightarrow True

cos (θ) (csc (θ) - sec (θ)) = cot (θ) - 1

tan^{4} (x) - sec^{4} (x) = 1 - 2 sec^{2} (x)

csc (x) + 1 = \frac{cot ( x )}{sec ( x ) - tan ( x )}

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

\frac{1 + sec ^{2} ( x )}{sec ^{2} ( x )} = 1 + cos^{2} (x)

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