Is cos^2(6θ)-sin^2(6θ)=cos(12θ) ?

The answer to whether cos^2(6θ)-sin^2(6θ)=cos(12θ) is True

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prove cos^2(6θ)-sin^2(6θ)=cos(12θ)

prove

cos^{2} (6 θ) - sin^{2} (6 θ) = cos (12 θ)

True

Solution steps

cos^{2} (6 θ) - sin^{2} (6 θ) = cos (12 θ)

Manipulating left side

cos^{2} (6 θ) - sin^{2} (6 θ)

Rewrite using trig identities

cos^{2} (6 θ) - sin^{2} (6 θ)

Use the Double Angle identity:

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

= cos (2 \cdot 6 θ)

Simplify

= cos (12 θ)

= cos (12 θ)

We showed that the two sides could take the same form

\Rightarrow True

6 cos^{4} (x) - 6 sin^{4} (x) = 6 cos (2 x)

sin (x) = sin (π - x)

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

tan (2 x) = \frac{2}{( cot ( x ) - tan ( x ) )}

sin (\frac{7 π}{12}) = sin (\frac{π}{3} + \frac{π}{4})

Is cos^2(6θ)-sin^2(6θ)=cos(12θ) ?
The answer to whether cos^2(6θ)-sin^2(6θ)=cos(12θ) is True