Is cos(2θ)=2cos^2(θ)-1 ?

The answer to whether cos(2θ)=2cos^2(θ)-1 is True

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prove cos(2θ)=2cos^2(θ)-1

prove

cos (2 θ) = 2 cos^{2} (θ) - 1

True

Solution steps

cos (2 θ) = 2 cos^{2} (θ) - 1

Manipulating left side

cos (2 θ)

Rewrite using trig identities

= cos^{2} (θ) - sin^{2} (θ)

Use the Pythagorean identity:

cos^{2} (θ) + sin^{2} (θ) = 1

sin^{2} (θ) = 1 - cos^{2} (θ)

= cos^{2} (θ) - (1 - cos^{2} (θ))

Simplify

cos^{2} (θ) - (1 - cos^{2} (θ)) : 2 cos^{2} (θ) - 1

= 2 cos^{2} (θ) - 1

We showed that the two sides could take the same form

\Rightarrow True

sin (\frac{7 π}{8})

sin (\frac{4 π}{3})

sin^{2} (x) = 3 cos^{2} (x)

sin (- π)

cos (5)