Is sec^2(θ)cot^2(θ)-1=cot^2(θ) ?

The answer to whether sec^2(θ)cot^2(θ)-1=cot^2(θ) is True

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prove sec^2(θ)cot^2(θ)-1=cot^2(θ)

Solution

prove

sec^{2} (θ) cot^{2} (θ) - 1 = cot^{2} (θ)

Solution

True

Solution steps

sec^{2} (θ) cot^{2} (θ) - 1 = cot^{2} (θ)

Manipulating left side

sec^{2} (θ) cot^{2} (θ) - 1

Express with sin, cos

- 1 + cot^{2} (θ) sec^{2} (θ)

Use the basic trigonometric identity:

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

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

(\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 )}

= - 1 + \frac{1}{sin ^{2} ( θ )}

Convert element to fraction:

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

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

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} ( θ ) + 1}{sin ^{2} ( θ )}

Multiply:

1 \cdot sin^{2} (θ) = sin^{2} (θ)

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

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

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

Manipulating right side

cot^{2} (θ)

Express with sin, cos

cot^{2} (θ)

Use the basic trigonometric identity:

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

= (\frac{cos ( θ )}{sin ( θ )})^{2}

Simplify

(\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{cos ^{2} ( θ )}{sin ^{2} ( θ )}

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

Rewrite using trig identities

\frac{cos ^{2} ( θ )}{sin ^{2} ( θ )}

Use the Pythagorean identity:

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

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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Frequently Asked Questions (FAQ)

Is sec^2(θ)cot^2(θ)-1=cot^2(θ) ?
The answer to whether sec^2(θ)cot^2(θ)-1=cot^2(θ) is True