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

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

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

prove

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

True

Solution steps

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

Manipulating left side

cos (3 t)

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Expand

(cos^{2} (t) - sin^{2} (t)) cos (t) - 2 sin^{2} (t) cos (t) : cos^{3} (t) - 3 sin^{2} (t) cos (t)

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

We showed that the two sides could take the same form

\Rightarrow True

sin^{2} (x) sec (x) csc (x) = tan (x)

4 sin (x) cos (x) + 1 = 2 sin (2 x) + 1

1 - \frac{tan ( θ ) cos ( θ )}{csc ( θ )} = cos^{2} (θ)

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

cot^{6} (x) = cot^{4} (x) csc^{2} (x) - cot^{4} (x)

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