Is 1+1/(tan^2(θ))= 1/(sin^2(θ)) ?

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

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prove 1+1/(tan^2(θ))= 1/(sin^2(θ))

prove

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

True

Solution steps

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

Manipulating left side

1 + \frac{1}{tan ^{2} ( θ )}

Express with sin, cos

1 + \frac{1}{tan ^{2} ( θ )}

Use the basic trigonometric identity:

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

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

Simplify

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

1 + \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 )}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply:

1 \cdot sin^{2} (θ) = sin^{2} (θ)

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

= \frac{1}{sin ^{2} ( θ )}

= \frac{1}{sin ^{2} ( θ )}

We showed that the two sides could take the same form

\Rightarrow True

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