Is cot(2θ)=(csc(θ)-2sin(θ))/(2cos(θ)) ?

The answer to whether cot(2θ)=(csc(θ)-2sin(θ))/(2cos(θ)) is True

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prove cot(2θ)=(csc(θ)-2sin(θ))/(2cos(θ))

prove

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

True

Solution steps

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

Manipulating right side

\frac{csc ( θ ) - 2 sin ( θ )}{2 cos ( θ )}

Express with sin, cos

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

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

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

csc (x) sin (x) = 1

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

Is cot(2θ)=(csc(θ)-2sin(θ))/(2cos(θ)) ?
The answer to whether cot(2θ)=(csc(θ)-2sin(θ))/(2cos(θ)) is True