Is (sin(x))/(1-cos^2(x))=csc(x) ?

The answer to whether (sin(x))/(1-cos^2(x))=csc(x) is True

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prove (sin(x))/(1-cos^2(x))=csc(x)

prove

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

True

Solution steps

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

Manipulating left side

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Cancel the common factor:

sin (x)

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{c s c ( x )}}

Simplify

\frac{1}{\frac{1}{c s c ( x )}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

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

Apply rule

\frac{a}{1} = a

= csc (x)

csc (x)

csc (x)

We showed that the two sides could take the same form

\Rightarrow True

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

cos (4 x) = cos^{2} (2 x) + sin^{2} (2 x)

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

csc (A) = \frac{cos ( 2 A )}{sin ( A )} + \frac{sin ( 2 A )}{cos ( A )}

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

Is (sin(x))/(1-cos^2(x))=csc(x) ?
The answer to whether (sin(x))/(1-cos^2(x))=csc(x) is True