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

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

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

prove

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

True

Solution steps

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

Manipulating left side

\frac{sin ( t )}{1 - cos ( t )}

Rewrite using trig identities

\frac{sin ( t )}{1 - cos ( t )}

Multiply by

\frac{1 + cos ( t )}{1 + cos ( t )}

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

Apply Difference of Two Squares Formula:

t^{2} - y^{2} = (t + y) (t - y)

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

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

Use the Pythagorean identity:

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

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

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

Simplify

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

sin (t)

= \frac{1 + cos ( t )}{sin ( t )}

= \frac{1 + cos ( t )}{sin ( t )}

= \frac{1 + cos ( t )}{sin ( t )}

Manipulating right side

csc (t) + cot (t)

Express with sin, cos

cot (t) + csc (t)

Use the basic trigonometric identity:

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

= \frac{cos ( t )}{sin ( t )} + csc (t)

Use the basic trigonometric identity:

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

= \frac{cos ( t )}{sin ( t )} + \frac{1}{sin ( t )}

Simplify

\frac{cos ( t )}{sin ( t )} + \frac{1}{sin ( t )} : \frac{cos ( t ) + 1}{sin ( t )}

\frac{cos ( t )}{sin ( t )} + \frac{1}{sin ( t )}

Apply rule

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

= \frac{cos ( t ) + 1}{sin ( t )}

= \frac{cos ( t ) + 1}{sin ( t )}

= \frac{1 + cos ( t )}{sin ( t )}

We showed that the two sides could take the same form

\Rightarrow True

sin (- x) + cot (- x) cos (- x) = - csc (x)

coth^{2} (x) - csch^{2} (x) = 1

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

csc (θ) (1 - cos^{2} (θ)) = sin (θ)

\frac{( csc ^{2} ( x ) )}{cot ( x )} = csc (x) sec (x)

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