Is cot(2θ)=(csc^2(θ)-2)/(2cot(θ)) ?

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

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prove cot(2θ)=(csc^2(θ)-2)/(2cot(θ))

prove

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

True

Solution steps

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

Manipulating right side

\frac{csc ^{2} ( θ ) - 2}{2 cot ( 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

tan (θ + \frac{π}{4}) = \frac{1 + tan ( θ )}{1 - tan ( θ )}

tan (θ) = \frac{1}{cot ( θ )}

cos (x) tan (x) sec (- x) = sin (x) - 1

sin (x) = cos (x - \frac{π}{2})

sin (θ) + cos (θ) = \frac{1 + tan ( θ )}{sec ( θ )}

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