Is cot(2x)=(csc^2(x)-2)/(2cot(x)) ?

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

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prove cot(2x)=(csc^2(x)-2)/(2cot(x))

prove

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

True

Solution steps

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

Manipulating right side

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

Express with sin, cos

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

Rewrite using trig identities

= \frac{cos ( 2 x )}{sin ( 2 x )}

Use the basic trigonometric identity:

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

= cot (2 x)

We showed that the two sides could take the same form

\Rightarrow True

(1 + csc (θ)) (1 - csc (θ)) = - cot^{2} (θ)

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

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

1 + tan (x) tan (\frac{x}{2}) = sec (x)

sin (x) \cdot tan (x) = sec (x) - cos (x)

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