Is cot^2(θ)+1= 1/(sin^2(θ)) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove cot^2(θ)+1= 1/(sin^2(θ))

prove

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

True

Solution steps

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

Manipulating left side

cot^{2} (θ) + 1

Express with sin, cos

1 + cot^{2} (θ)

Use the basic trigonometric identity:

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

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

Simplify

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

1 + (\frac{cos ( θ )}{sin ( θ )})^{2}

Apply exponent rule:

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

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot sin^{2} (θ) = sin^{2} (θ)

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= \frac{1}{sin ^{2} ( θ )}

= \frac{1}{sin ^{2} ( θ )}

We showed that the two sides could take the same form

\Rightarrow True

3 cos (2 x + \frac{π}{2}) - 2 = 6 cos^{2} (x + \frac{π}{4})

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

cot^{4} (x) = \frac{cos ^{4} ( x )}{sin ^{4} ( x )}

\frac{1}{1 - sin ( x )} + \frac{1}{1 + sin ( x )} = 2 sec (x)

cos^{4} (b) - sin^{4} (b) = cos (2 b)

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