Is (1+cos(2x))/(sin^2(x))=2cot^2(x) ?

The answer to whether (1+cos(2x))/(sin^2(x))=2cot^2(x) is True

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prove (1+cos(2x))/(sin^2(x))=2cot^2(x)

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

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

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

Group like terms

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

1 - 1 = 0

= 2 cos^{2} (x)

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

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

Use the basic trigonometric identity:

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

2 cot (x) cot (x)

Simplify

2 cot (x) cot (x) : 2 cot^{2} (x)

2 cot (x) cot (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cot (x) cot (x) = cot^{1 + 1} (x)

= 2 cot^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= 2 cot^{2} (x)

2 cot^{2} (x)

2 cot^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

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

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

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

2 sin (2 x) = 4 sin (x) cos (x)

(sin (x)) (csc (x) - sin (x)) = cos^{2} (x)

Is (1+cos(2x))/(sin^2(x))=2cot^2(x) ?
The answer to whether (1+cos(2x))/(sin^2(x))=2cot^2(x) is True