Is csc^2(x)cos^2(x)= 1/(tan^2(x)) ?

The answer to whether csc^2(x)cos^2(x)= 1/(tan^2(x)) is True

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prove csc^2(x)cos^2(x)= 1/(tan^2(x))

prove

csc^{2} (x) cos^{2} (x) = \frac{1}{tan ^{2} ( x )}

True

Solution steps

csc^{2} (x) cos^{2} (x) = \frac{1}{tan ^{2} ( x )}

Manipulating right side

\frac{1}{tan ^{2} ( x )}

Express with sin, cos

\frac{1}{tan ^{2} ( x )}

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Apply exponent rule:

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

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

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

Apply the fraction rule:

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

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

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

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

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{cos ^{2} ( x )}{( \frac{1}{c s c ( x )} ) ^{2}}

Simplify

\frac{cos ^{2} ( x )}{( \frac{1}{c s c ( x )} ) ^{2}}

(\frac{1}{csc ( x )})^{2} = \frac{1}{csc ^{2} ( x )}

(\frac{1}{csc ( x )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{csc ^{2} ( x )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{csc ^{2} ( x )}

= \frac{cos ^{2} ( x )}{\frac{1}{c s c ^{2} ( x )}}

Apply the fraction rule:

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

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

Apply rule

\frac{a}{1} = a

= cos^{2} (x) csc^{2} (x)

cos^{2} (x) csc^{2} (x)

cos^{2} (x) csc^{2} (x)

= csc^{2} (x) cos^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

cot (t) tan (t) = 1

1 + tan^{2} (t) = sec^{2} (t)

sec^{2} (x) = \frac{csc ( x )}{csc ( x ) - sin ( x )}

\frac{cos ( θ )}{sin ( θ )} = cot (θ)

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

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