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

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

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

prove

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

True

Solution steps

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

Manipulating left side

\frac{sin ( θ )}{1 - cos ( θ )}

Rewrite using trig identities

\frac{sin ( θ )}{1 - cos ( θ )}

Multiply by

\frac{1 + cos ( θ )}{1 + cos ( θ )}

= \frac{sin ( θ )}{\frac{( 1 - c o s ( θ ) ) ( 1 + c o s ( θ ) )}{1 + c o s ( θ )}}

Apply Difference of Two Squares Formula:

θ^{2} - y^{2} = (θ + y) (θ - y)

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

= \frac{sin ( θ )}{\frac{1 - c o s ^{2} ( θ )}{1 + c o s ( θ )}}

Use the Pythagorean identity:

1 = cos^{2} (θ) + sin^{2} (θ)

1 - cos^{2} (θ) = sin^{2} (θ)

= \frac{sin ( θ )}{\frac{s i n ^{2} ( θ )}{1 + c o s ( θ )}}

Simplify

\frac{sin ( θ )}{\frac{s i n ^{2} ( θ )}{1 + c o s ( θ )}} : \frac{1 + cos ( θ )}{sin ( θ )}

\frac{sin ( θ )}{\frac{s i n ^{2} ( θ )}{1 + c o s ( θ )}}

Apply the fraction rule:

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

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

Cancel the common factor:

sin (θ)

= \frac{1 + cos ( θ )}{sin ( θ )}

= \frac{1 + cos ( θ )}{sin ( θ )}

= \frac{1 + cos ( θ )}{sin ( 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

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