Is cos^4(x)-sin^4(x)=2cos^2(x)-1 ?

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

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prove cos^4(x)-sin^4(x)=2cos^2(x)-1

prove

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

True

Solution steps

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

Manipulating left side

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

Factor

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

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

Rewrite using trig identities

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

We showed that the two sides could take the same form

\Rightarrow True

tan (x) (cot (x) + tan (x)) = sec^{2} (x)

\frac{1 + sin ( 2 x )}{sin ( 2 x )} = 1 + \frac{1}{2} sec (x) csc (x)

(1 - sin^{2} (x)) (1 - sec^{2} (x)) = - sin^{2} (x)

\frac{1}{csc ( x ) + 1} + \frac{1}{sin ( x ) + 1} = 1

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

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