Is (1+cos(t))/(1-cos(t))=(csc(t)+cot(t))^2 ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove (1+cos(t))/(1-cos(t))=(csc(t)+cot(t))^2

prove

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

True

Solution steps

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

Manipulating right side

(csc (t) + cot (t))^{2}

Express with sin, cos

(cot (t) + csc (t))^{2}

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

Combine the fractions

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

Apply exponent rule:

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Factor

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

1 - cos^{2} (t)

Apply Difference of Two Squares Formula:

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

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

= (1 + cos (t)) (1 - cos (t))

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

Cancel the common factor:

cos (t) + 1

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

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

We showed that the two sides could take the same form

\Rightarrow True

sec^{2} (α) + csc^{2} (α) = \frac{1}{sin ^{2} ( α ) cos ^{2} ( α )}

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

tan (x) = sin (x) sec (x)

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

csc (2 x) + cot (2 x) = cot (x)

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