Is 1/(sin^2(θ))-1/(tan^2(θ))=1 ?

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

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prove 1/(sin^2(θ))-1/(tan^2(θ))=1

prove

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

True

Solution steps

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

Manipulating left side

\frac{1}{sin ^{2} ( θ )} - \frac{1}{tan ^{2} ( θ )}

Express with sin, cos

\frac{1}{sin ^{2} ( θ )} - \frac{1}{tan ^{2} ( θ )}

Use the basic trigonometric identity:

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

= \frac{1}{sin ^{2} ( θ )} - \frac{1}{( \frac{s i n ( θ )}{c o s ( θ )} ) ^{2}}

Simplify

\frac{1}{sin ^{2} ( θ )} - \frac{1}{( \frac{s i n ( θ )}{c o s ( θ )} ) ^{2}} : \frac{1 - cos ^{2} ( θ )}{sin ^{2} ( θ )}

\frac{1}{sin ^{2} ( θ )} - \frac{1}{( \frac{s i n ( θ )}{c o s ( θ )} ) ^{2}}

\frac{1}{( \frac{s i n ( θ )}{c o s ( θ )} ) ^{2}} = \frac{cos ^{2} ( θ )}{sin ^{2} ( θ )}

\frac{1}{( \frac{s i n ( θ )}{c o s ( θ )} ) ^{2}}

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

(\frac{sin ( θ )}{cos ( θ )})^{2}

Apply exponent rule:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

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

= \frac{1}{\frac{s i n ^{2} ( θ )}{c o s ^{2} ( θ )}}

Apply the fraction rule:

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

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

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

Apply rule

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Apply rule

\frac{a}{a} = 1

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

(5 sin (θ) - 5 sin (θ))^{2} = 25 - 25 sin (2 θ)

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

\frac{1}{( sec ( y ) - tan ( y ) )} = sec (y) + tan (y)

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

2 cos (2 x) = 4 cos^{2} (x) - 2

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