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

The answer to whether sin(x)(csc(x)-sin(x))=cos^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 sin(x)(csc(x)-sin(x))=cos^2(x)

prove

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

True

Solution steps

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

Manipulating left side

sin (x) (csc (x) - sin (x))

Express with sin, cos

(csc (x) - sin (x)) sin (x)

Use the basic trigonometric identity:

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

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

Simplify

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

(\frac{1}{sin ( x )} - sin (x)) sin (x)

Join

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

\frac{1}{sin ( x )} - sin (x)

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

1 - sin (x) sin (x)

sin (x) sin (x) = sin^{2} (x)

sin (x) sin (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

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

= sin^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= sin^{2} (x)

= 1 - sin^{2} (x)

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin (x)

= 1 - sin^{2} (x)

= 1 - sin^{2} (x)

= 1 - sin^{2} (x)

Rewrite using trig identities

1 - sin^{2} (x)

Use the Pythagorean identity:

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

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

= cos^{2} (x)

= cos^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

cot (x) - tan (x) = sec (x) (csc (x) - 2 sin (x))

1 + cot^{2} (- θ) = csc^{2} (θ)

sec^{2} (x) = 1 + tan^{2} (x)

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

cos (x) (1 + tan^{2} (x)) = sec (x)

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