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

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

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

prove

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

True

Solution steps

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

Manipulating left side

cos^{4} (x) - sin^{4} (x) + 1

Factor

1 - sin^{4} (x) : - (sin^{2} (x) + 1) (sin (x) + 1) (sin (x) - 1)

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

Rewrite using trig identities

= 2 cos^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

sin (x) cos (x) = \frac{sin ( 2 x )}{2}

- tan (y) + sec (y) = \frac{cos ( y )}{1 + sin ( y )}

1 - \frac{cos ^{2} ( θ )}{1 - sin ( θ )} = - sin (θ)

sin (2 z) = 2 sin (z) cos (z)

csc^{2} (a) sec^{2} (a) = sec^{2} (a) + csc^{2} (a)

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