Is 4cos^3(x)-3cos(x)=cos(3x) ?

The answer to whether 4cos^3(x)-3cos(x)=cos(3x) is True

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prove 4cos^3(x)-3cos(x)=cos(3x)

prove

4 cos^{3} (x) - 3 cos (x) = cos (3 x)

True

Solution steps

4 cos^{3} (x) - 3 cos (x) = cos (3 x)

Manipulating right side

cos (3 x)

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Use the Pythagorean identity:

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

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

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

Expand

(2 cos^{2} (x) - 1) cos (x) - 2 (1 - cos^{2} (x)) cos (x) : 4 cos^{3} (x) - 3 cos (x)

= 4 cos^{3} (x) - 3 cos (x)

We showed that the two sides could take the same form

\Rightarrow True

\frac{csc ^{2} ( θ )}{cot ( θ )} = sec (θ) csc (θ)

tan (θ) cot (θ) - cos^{2} (θ) = sin^{2} (θ)

\frac{cot ^{2} ( x )}{csc ( x ) + 1} = csc (x) - 1

csc^{2} (- θ) - 1 = cot^{2} (θ)

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

Is 4cos^3(x)-3cos(x)=cos(3x) ?
The answer to whether 4cos^3(x)-3cos(x)=cos(3x) is True