Is sec^2(θ)cot^2(θ)=csc^2(θ) ?

The answer to whether sec^2(θ)cot^2(θ)=csc^2(θ) is True

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prove sec^2(θ)cot^2(θ)=csc^2(θ)

prove

sec^{2} (θ) cot^{2} (θ) = csc^{2} (θ)

True

Solution steps

sec^{2} (θ) cot^{2} (θ) = csc^{2} (θ)

Manipulating left side

sec^{2} (θ) cot^{2} (θ)

Express with sin, cos

cot^{2} (θ) sec^{2} (θ)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

sec (x) = \frac{1}{cos ( x )}

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

Simplify

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

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

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

(\frac{cos ( θ )}{sin ( θ )})^{2}

Apply exponent rule:

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

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

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

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

(\frac{1}{cos ( θ )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{cos ^{2} ( θ )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{cos ^{2} ( θ )}

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

Multiply fractions:

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

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

Cancel the common factor:

cos^{2} (θ)

= \frac{1}{sin ^{2} ( θ )}

= \frac{1}{sin ^{2} ( θ )}

= \frac{1}{sin ^{2} ( θ )}

Rewrite using trig identities

Use the basic trigonometric identity:

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

\frac{1}{( \frac{1}{c s c ( θ )} ) ^{2}}

Simplify

\frac{1}{( \frac{1}{c s c ( θ )} ) ^{2}}

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

(\frac{1}{csc ( θ )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{csc ^{2} ( θ )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{csc ^{2} ( θ )}

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

Apply the fraction rule:

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

= \frac{csc ^{2} ( θ )}{1}

Apply rule

\frac{a}{1} = a

= csc^{2} (θ)

csc^{2} (θ)

csc^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True

tan (\frac{3 π}{2} - x) = cot (x)

\frac{3 csc ( - x )}{sec ( - x )} = - 3 cot (x)

\frac{1 - tan ^{4} ( x )}{1 - tan ^{2} ( x )} = sec^{2} (x)

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

cos (4 x) = 1 - 2 sin^{2} (2 x)

Is sec^2(θ)cot^2(θ)=csc^2(θ) ?
The answer to whether sec^2(θ)cot^2(θ)=csc^2(θ) is True