Is (1+cot^2(x))= 1/(sin^2(x)) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

prove (1+cot^2(x))= 1/(sin^2(x))

prove

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

True

Solution steps

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

Manipulating left side

1 + cot^{2} (x)

Express with sin, cos

1 + cot^{2} (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply exponent rule:

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

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

Convert element to fraction:

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

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

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

Multiply:

1 \cdot sin^{2} (x) = sin^{2} (x)

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

sin (x) (1 + csc (x)) = 1 + sin (x)

(sec^{2} (θ) - 1) (sin^{2} (θ) - 1) = - sin^{2} (θ)

\frac{1}{tan ( x )} = csc (x) cos (x)

csc^{2} (x) (1 - cos^{2} (x)) = sin (648 0^{\circ})

- 2 cos^{4} (x) = - 2 (cos^{2} (x)) (cos^{2} (x))

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