Is sin(u)csc(u)-cos^2(u)=sin^2(u) ?

The answer to whether sin(u)csc(u)-cos^2(u)=sin^2(u) is True

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

prove sin(u)csc(u)-cos^2(u)=sin^2(u)

prove

sin (u) csc (u) - cos^{2} (u) = sin^{2} (u)

True

Solution steps

sin (u) csc (u) - cos^{2} (u) = sin^{2} (u)

Manipulating left side

sin (u) csc (u) - cos^{2} (u)

Express with sin, cos

- cos^{2} (u) + csc (u) sin (u)

Use the basic trigonometric identity:

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

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

Simplify

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

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

\frac{1}{sin ( u )} sin (u) = 1

\frac{1}{sin ( u )} sin (u)

Multiply fractions:

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

= \frac{1 \cdot sin ( u )}{sin ( u )}

Cancel the common factor:

sin (u)

= 1

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

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

= 1 - cos^{2} (u)

Rewrite using trig identities

1 - cos^{2} (u)

Use the Pythagorean identity:

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

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

= sin^{2} (u)

= sin^{2} (u)

We showed that the two sides could take the same form

\Rightarrow True

Is sin(u)csc(u)-cos^2(u)=sin^2(u) ?
The answer to whether sin(u)csc(u)-cos^2(u)=sin^2(u) is True