Is 1/(1-cos^2(θ))=1+cot^2(θ) ?

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

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prove 1/(1-cos^2(θ))=1+cot^2(θ)

prove

\frac{1}{1 - cos ^{2} ( θ )} = 1 + cot^{2} (θ)

True

Solution steps

\frac{1}{1 - cos ^{2} ( θ )} = 1 + cot^{2} (θ)

Manipulating right side

1 + cot^{2} (θ)

Express with sin, cos

1 + cot^{2} (θ)

Use the basic trigonometric identity:

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

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

Simplify

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

1 + (\frac{cos ( θ )}{sin ( θ )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

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

Multiply:

1 \cdot sin^{2} (θ) = sin^{2} (θ)

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

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

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

Rewrite using trig identities

\frac{cos ^{2} ( θ ) + sin ^{2} ( θ )}{sin ^{2} ( θ )}

Use the Pythagorean identity:

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

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

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

cos^{2} (θ) + 1 - cos^{2} (θ) = 1

cos^{2} (θ) + 1 - cos^{2} (θ)

Group like terms

= cos^{2} (θ) - cos^{2} (θ) + 1

Add similar elements:

cos^{2} (θ) - cos^{2} (θ) = 0

= 1

= \frac{1}{1 - cos ^{2} ( θ )}

= \frac{1}{1 - cos ^{2} ( θ )}

We showed that the two sides could take the same form

\Rightarrow True

sec (θ + π) = - sec (θ)

sec (2 x) = \frac{sec ^{2} ( x ) + sec ^{4} ( x )}{2 + sec ^{2} ( x ) - sec ^{4} ( x )}

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

sin (θ) - cos (θ) = 1

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

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