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

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

cos (θ) + tan (θ) sin (θ) = sec (θ)

cot^{2} (x) + 5 = csc^{2} (x) + 4

\frac{sin ( θ )}{sin ( θ ) - 1} = \frac{1}{1 - csc ( θ )}

cot (x) (sec (x) - cos (x)) = sin (x)

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