Is cos(3β)=cos(β)(cos^2(β)-3sin^2(β)) ?

The answer to whether cos(3β)=cos(β)(cos^2(β)-3sin^2(β)) is True

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prove cos(3β)=cos(β)(cos^2(β)-3sin^2(β))

prove

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

True

Solution steps

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

Manipulating left side

cos (3 β)

Rewrite using trig identities

= 4 cos^{3} (β) - 3 cos (β)

Factor

- 3 cos (β) + 4 cos^{3} (β) : cos (β) (- 3 + 4 cos^{2} (β))

= (- 3 + 4 cos^{2} (β)) cos (β)

Rewrite using trig identities

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

We showed that the two sides could take the same form

\Rightarrow True

sec^{2} (θ) cot^{2} (θ) - 1 = cot^{2} (θ)

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

cot (a) = \frac{1}{tan ( a )}

cos (u) sec (u) - cos^{2} (u) = sin^{2} (u)

\frac{sec ( θ )}{cot ( θ ) + tan ( θ )} = sin (θ)

Is cos(3β)=cos(β)(cos^2(β)-3sin^2(β)) ?
The answer to whether cos(3β)=cos(β)(cos^2(β)-3sin^2(β)) is True