Is csc^2(x)*cos^2(x)=cot^2(x) ?

The answer to whether csc^2(x)*cos^2(x)=cot^2(x) is True

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prove csc^2(x)*cos^2(x)=cot^2(x)

prove

csc^{2} (x) \cdot cos^{2} (x) = cot^{2} (x)

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Multiply fractions:

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

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

Multiply:

1 \cdot cos^{2} (x) = cos^{2} (x)

= \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

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

Use the basic trigonometric identity:

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

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

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

Use the basic trigonometric identity:

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

cot (x) cot (x)

Simplify

cot (x) cot (x) : cot^{2} (x)

cot (x) cot (x)

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

cot (x) cot (x) = cot^{1 + 1} (x)

= cot^{1 + 1} (x)

Add the numbers:

1 + 1 = 2

= cot^{2} (x)

cot^{2} (x)

cot^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

\frac{1}{sec ^{3} ( x ) cos ^{4} ( x )} = sec (x)

csc^{2} (θ) + 1 = cot^{2} (θ)

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

- sin^{2} (θ) = cos^{2} (θ)

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

Is csc^2(x)*cos^2(x)=cot^2(x) ?
The answer to whether csc^2(x)*cos^2(x)=cot^2(x) is True