Is prove-(-1+sin^2(θ))/(sin^2(θ))=cot^2(θ) ?

The answer to whether prove-(-1+sin^2(θ))/(sin^2(θ))=cot^2(θ) is True

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prove-(-1+sin^2(θ))/(sin^2(θ))=cot^2(θ)

Solution

prove

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

Solution

True

Solution steps

- \frac{- 1 + sin ^{2} ( θ )}{sin ^{2} ( θ )} = cot^{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 Double Angle identity:

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

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

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

Use the Pythagorean identity:

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

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

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

Use the Double Angle identity:

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

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

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

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

Group like terms

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

Add similar elements:

cos^{2} (θ) - cos^{2} (θ) = 0

= - sin^{2} (θ) + 1

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

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

Factor

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

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

Factor

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

1 - cos^{2} (θ)

Factor out common term

- 1

= - (cos^{2} (θ) - 1)

Factor

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

cos^{2} (θ) - 1

Rewrite

1

1^{2}

= cos^{2} (θ) - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

= (cos (θ) + 1) (cos (θ) - 1)

= - (cos (θ) + 1) (cos (θ) - 1)

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

Factor

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

1 - sin^{2} (θ)

Factor out common term

- 1

= - (sin^{2} (θ) - 1)

Factor

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

sin^{2} (θ) - 1

Rewrite

1

1^{2}

= sin^{2} (θ) - 1^{2}

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

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

= (sin (θ) + 1) (sin (θ) - 1)

= - (sin (θ) + 1) (sin (θ) - 1)

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

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

Expand

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

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

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

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

Expand

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

(- 1 + cos (θ)) (1 + cos (θ))

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = cos (θ), b = 1

= cos^{2} (θ) - 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= cos^{2} (θ) - 1

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

Expand

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

(- 1 + sin (θ)) (1 + sin (θ))

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = sin (θ), b = 1

= sin^{2} (θ) - 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= sin^{2} (θ) - 1

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Simplify

= - \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 prove-(-1+sin^2(θ))/(sin^2(θ))=cot^2(θ) ?
The answer to whether prove-(-1+sin^2(θ))/(sin^2(θ))=cot^2(θ) is True