Is (1-cos(θ))/(sin(θ))=csc(θ)-cot(θ) ?

The answer to whether (1-cos(θ))/(sin(θ))=csc(θ)-cot(θ) is True

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prove (1-cos(θ))/(sin(θ))=csc(θ)-cot(θ)

prove

\frac{1 - cos ( θ )}{sin ( θ )} = csc (θ) - cot (θ)

True

Solution steps

\frac{1 - cos ( θ )}{sin ( θ )} = csc (θ) - cot (θ)

Manipulating right side

csc (θ) - cot (θ)

Express with sin, cos

- cot (θ) + csc (θ)

Use the basic trigonometric identity:

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

= - \frac{cos ( θ )}{sin ( θ )} + csc (θ)

Use the basic trigonometric identity:

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

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

Simplify

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

- \frac{cos ( θ )}{sin ( θ )} + \frac{1}{sin ( θ )}

Apply rule

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

= \frac{- cos ( θ ) + 1}{sin ( θ )}

= \frac{- cos ( θ ) + 1}{sin ( θ )}

= \frac{1 - cos ( θ )}{sin ( θ )}

We showed that the two sides could take the same form

\Rightarrow True

csc^{2} (x) (1 - sin^{2} (x)) = cot^{2} (x)

sin (3 x) = (sin (x)) (4 cos^{2} (x) - 1)

sin (2 a) = 2 sin (a) cos (a)

tan (x + \frac{π}{2}) = - cot (x)

cos (θ) tan (θ) csc (θ) = 1

Is (1-cos(θ))/(sin(θ))=csc(θ)-cot(θ) ?
The answer to whether (1-cos(θ))/(sin(θ))=csc(θ)-cot(θ) is True