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

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

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

prove

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

True

Solution steps

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

Manipulating right side

\frac{1}{2} cot (x) - \frac{1}{2} tan (x)

Express with sin, cos

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

(sin (t) - cos (t))^{2} = 1 - sin (2 t)

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

sin (θ) = \frac{2 tan ( \frac{θ}{2} )}{1 + tan ^{2} ( \frac{θ}{2} )}

sin^{2} (x) - 2 cos^{2} (x) = 1 - 3 cos^{2} (x)

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

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