Is 1+cos(x)=2cos^2(x/2) ?

The answer to whether 1+cos(x)=2cos^2(x/2) is True

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prove 1+cos(x)=2cos^2(x/2)

prove

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

True

Solution steps

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

Let:

u = \frac{x}{2}

1 + cos (2 u) = 2 cos^{2} (u)

Prove

1 + cos (2 u) = 2 cos^{2} (u) :

True

Therefore

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

We showed that the two sides could take the same form

\Rightarrow True

\frac{csc ^{4} ( x ) - 1}{1 + csc ^{2} ( x )} = cot^{2} (x)

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

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

sin^{2} (θ) + sin^{2} (θ) tan^{2} (θ) = tan^{2} (θ)

1 + sin (2 θ) = (cos (θ) sin (θ))^{2}