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

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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

tan (x) + cot (x) - sec (x) csc (x) = 0

\frac{1 - sin ( x )}{sec ( x )} = \frac{cos ^{3} ( x )}{1 + sin ( x )}

(1 + cos (θ)) (csc (θ) - cot (θ)) = sin (θ)

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

(1 + sec (x)) (csc (x) - cot (x)) = tan (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