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

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

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prove 1/(cos^2(x))-1/(cot^2(x))=1

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

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

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

Apply exponent rule:

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

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

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

Apply the fraction rule:

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

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

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

Apply rule

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

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Apply rule

\frac{a}{a} = 1

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

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

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

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

sin (0) = (0)^{3}

tan (2 x) = - \frac{arctan ( - \frac{4}{3} )}{2}

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