Is csc^2(θ)= 1/(sin^2(θ)) ?

The answer to whether csc^2(θ)= 1/(sin^2(θ)) is True

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prove csc^2(θ)= 1/(sin^2(θ))

prove

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

True

Solution steps

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

Manipulating left side

csc^{2} (θ)

Express with sin, cos

csc^{2} (θ)

Use the basic trigonometric identity:

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

= (\frac{1}{sin ( θ )})^{2}

Simplify

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

(\frac{1}{sin ( θ )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{sin ^{2} ( θ )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( θ )}

= \frac{1}{sin ^{2} ( θ )}

= \frac{1}{sin ^{2} ( θ )}

We showed that the two sides could take the same form

\Rightarrow True

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

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

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

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

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

Is csc^2(θ)= 1/(sin^2(θ)) ?
The answer to whether csc^2(θ)= 1/(sin^2(θ)) is True