Is ((1))/((2))(cot(θ)-tan(θ))=cot(2θ) ?

The answer to whether ((1))/((2))(cot(θ)-tan(θ))=cot(2θ) is True

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prove ((1))/((2))(cot(θ)-tan(θ))=cot(2θ)

prove

\frac{( 1 )}{( 2 )} (cot (θ) - tan (θ)) = cot (2 θ)

True

Solution steps

\frac{1}{2} (cot (θ) - tan (θ)) = cot (2 θ)

Manipulating left side

\frac{1}{2} (cot (θ) - tan (θ))

Express with sin, cos

= \frac{cos ^{2} ( θ ) - sin ^{2} ( θ )}{2 cos ( θ ) sin ( θ )}

Rewrite using trig identities

= \frac{cos ( 2 θ )}{sin ( 2 θ )}

Use the basic trigonometric identity:

\frac{cos ( x )}{sin ( x )} = cot (x)

= cot (2 θ)

We showed that the two sides could take the same form

\Rightarrow True

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

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

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

3 sec^{2} (x) - 4 = 0

sin (3 θ) = 3 sin (θ) - 4 sin (3 θ)

Is ((1))/((2))(cot(θ)-tan(θ))=cot(2θ) ?
The answer to whether ((1))/((2))(cot(θ)-tan(θ))=cot(2θ) is True