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

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

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prove csc(x)-(cos^2(x))/(sin(x))=sin(x)

prove

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

True

Solution steps

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

Manipulating left side

csc (x) - \frac{cos ^{2} ( x )}{sin ( x )}

Express with sin, cos

csc (x) - \frac{cos ^{2} ( x )}{sin ( x )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

Apply rule

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Cancel the common factor:

sin (x)

= sin (x)

= sin (x)

We showed that the two sides could take the same form

\Rightarrow True

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

cos^{2} (A) tan^{2} (A) = sin^{2} (A)

(1 - cos (x)) (1 + sec (x)) = sin (x)

csc (y) cos^{2} (y) + sin (y) = csc (y)

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

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