Is cos(8θ)=cos^2(4θ)-sin^2(4θ) ?

The answer to whether cos(8θ)=cos^2(4θ)-sin^2(4θ) is True

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prove cos(8θ)=cos^2(4θ)-sin^2(4θ)

prove

cos (8 θ) = cos^{2} (4 θ) - sin^{2} (4 θ)

True

Solution steps

cos (8 θ) = cos^{2} (4 θ) - sin^{2} (4 θ)

Manipulating right side

cos^{2} (4 θ) - sin^{2} (4 θ)

Rewrite using trig identities

cos^{2} (4 θ) - sin^{2} (4 θ)

Use the Double Angle identity:

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

= cos (2 \cdot 4 θ)

Simplify

= cos (8 θ)

= cos (8 θ)

We showed that the two sides could take the same form

\Rightarrow True

2 sec (2 α) cos (2 α) = 2

\frac{1}{sec ( x )} + 1 = \frac{sec ( x ) + 1}{sec ( x )}

tan^{2} (θ) = sec (θ) csc (θ) tan (θ) - 1

sin (x + y) = cos (\frac{π}{2} - (x + y))

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

Is cos(8θ)=cos^2(4θ)-sin^2(4θ) ?
The answer to whether cos(8θ)=cos^2(4θ)-sin^2(4θ) is True