Is 1+1/(tan^2(a))= 1/(sin^2(a)) ?

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

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prove 1+1/(tan^2(a))= 1/(sin^2(a))

prove

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

True

Solution steps

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

Manipulating left side

1 + \frac{1}{tan ^{2} ( a )}

Express with sin, cos

1 + \frac{1}{tan ^{2} ( a )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

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

(\frac{sin ( a )}{cos ( a )})^{2}

Apply exponent rule:

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

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

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

Apply the fraction rule:

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

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

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

Convert element to fraction:

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

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

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} ( a ) + cos ^{2} ( a )}{sin ^{2} ( a )}

Multiply:

1 \cdot sin^{2} (a) = sin^{2} (a)

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

\frac{tan ( θ )}{cot ( θ )} = tan^{2} (θ)

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

\frac{sin ( 6 x )}{cos ( x )} = 6 sin (x)

csc (x) sec (x) - tan (x) = cot (x)

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

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