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

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

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

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Double Angle identity:

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

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

Simplify

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

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

Divide the numbers:

\frac{2}{2} = 1

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

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

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

Rewrite using trig identities

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

Use the basic trigonometric identity:

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

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

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

Use the basic trigonometric identity:

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

cot (x) cot (x)

Simplify

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

cot (x) cot (x)

Apply exponent rule:

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

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

= cot^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cot^{2} (x)

cot^{2} (x)

cot^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

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