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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Factor

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

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

Rewrite using trig identities

= 2 cos^{2} (A)

We showed that the two sides could take the same form

\Rightarrow True

- cot (θ) cos (θ) - sin (θ) = - cot (θ) sec (θ)

cot (θ) = \frac{1}{\frac{s i n ( θ )}{c o s ( θ )}}

\frac{cos ( x ) + cot ( x )}{csc ( x ) + 1} = cos (x)

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

1 = 1 + cot (θ)

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