Is cos(2a)=2cos^2(a)-1 ?

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

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prove cos(2a)=2cos^2(a)-1

prove

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

True

Solution steps

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

Manipulating left side

cos (2 a)

Rewrite using trig identities

= cos^{2} (a) - sin^{2} (a)

Use the Pythagorean identity:

cos^{2} (a) + sin^{2} (a) = 1

sin^{2} (a) = 1 - cos^{2} (a)

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

Simplify

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

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

We showed that the two sides could take the same form

\Rightarrow True

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