Is (csc^2(θ)-1)/(csc^2(θ))=cos^2(θ) ?

The answer to whether (csc^2(θ)-1)/(csc^2(θ))=cos^2(θ) is True

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prove (csc^2(θ)-1)/(csc^2(θ))=cos^2(θ)

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

= \frac{- 1 + ( \frac{1}{s i n ( θ )} ) ^{2}}{( \frac{1}{s i n ( θ )} ) ^{2}}

Simplify

\frac{- 1 + ( \frac{1}{s i n ( θ )} ) ^{2}}{( \frac{1}{s i n ( θ )} ) ^{2}} : - sin^{2} (θ) + 1

\frac{- 1 + ( \frac{1}{s i n ( θ )} ) ^{2}}{( \frac{1}{s i n ( θ )} ) ^{2}}

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

(\frac{1}{sin ( θ )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{sin ^{2} ( θ )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( θ )}

= \frac{- 1 + ( \frac{1}{s i n ( θ )} ) ^{2}}{\frac{1}{s i n ^{2} ( θ )}}

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

(\frac{1}{sin ( θ )})^{2}

Apply exponent rule:

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

= \frac{1 ^{2}}{sin ^{2} ( θ )}

Apply rule

1^{a} = 1

1^{2} = 1

= \frac{1}{sin ^{2} ( θ )}

= \frac{- 1 + \frac{1}{s i n ^{2} ( θ )}}{\frac{1}{s i n ^{2} ( θ )}}

Apply the fraction rule:

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

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

Join

- 1 + \frac{1}{sin ^{2} ( θ )} : \frac{- sin ^{2} ( θ ) + 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{\frac{- s i n ^{2} ( θ ) + 1}{s i n ^{2} ( θ )} sin ^{2} ( θ )}{1}

Apply the fraction rule:

\frac{a}{1} = a

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

Multiply fractions:

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

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

Cancel the common factor:

sin^{2} (θ)

= - - sin^{2} (θ) + 1

= - sin^{2} (θ) + 1

= 1 - sin^{2} (θ)

Rewrite using trig identities

1 - sin^{2} (θ)

Use the Pythagorean identity:

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

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

= cos^{2} (θ)

= cos^{2} (θ)

We showed that the two sides could take the same form

\Rightarrow True

1 + cos (x) \cdot sin (x) = sin (x)

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

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

cos^{2} (2 x) = \frac{( 1 + cos ( 4 x ) )}{2}

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

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