Is csc^2(θ)(1-cos^2(θ))=1 ?

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

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prove csc^2(θ)(1-cos^2(θ))=1

prove

csc^{2} (θ) (1 - cos^{2} (θ)) = 1

True

Solution steps

csc^{2} (θ) (1 - cos^{2} (θ)) = 1

Manipulating left side

csc^{2} (θ) (1 - cos^{2} (θ))

Express with sin, cos

(1 - cos^{2} (θ)) csc^{2} (θ)

Use the basic trigonometric identity:

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

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

Simplify

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

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

Multiply fractions:

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

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

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

1 \cdot (1 - cos^{2} (θ))

Multiply:

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

= (1 - cos^{2} (θ))

Remove parentheses:

(a) = a

= 1 - cos^{2} (θ)

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

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

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

Rewrite using trig identities

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

Use the Pythagorean identity:

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

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

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

Apply rule

\frac{a}{a} = 1

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

cos^{2} (θ) + sin^{2} (θ) = 1

sin (\frac{3 π}{2} + θ) = - cos (θ)

(1 - cos^{2} (x)) (1 + tan^{2} (x)) = tan^{2} (x)

2 cot (2 x) = cot (x) - tan (x)

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

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