Is (sin(θ))/(1-cos^2(θ))=csc(θ) ?

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

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prove (sin(θ))/(1-cos^2(θ))=csc(θ)

prove

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

True

Solution steps

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

Manipulating left side

\frac{sin ( θ )}{1 - cos ^{2} ( θ )}

Rewrite using trig identities

\frac{sin ( θ )}{1 - cos ^{2} ( θ )}

Use the Pythagorean identity:

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

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

= \frac{sin ( θ )}{sin ^{2} ( θ )}

Cancel the common factor:

sin (θ)

= \frac{1}{sin ( θ )}

= \frac{1}{sin ( θ )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{\frac{1}{c s c ( θ )}}

Simplify

\frac{1}{\frac{1}{c s c ( θ )}}

Apply the fraction rule:

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

= \frac{csc ( θ )}{1}

Apply rule

\frac{a}{1} = a

= csc (θ)

csc (θ)

csc (θ)

We showed that the two sides could take the same form

\Rightarrow True

sin (a) sec (a) = tan (a)

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

cos^{2} (θ) + cos^{2} (θ) cot^{2} (θ) = cot^{2} (θ)

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

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

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