Is sin^2(x)(csc^2(x)-1)=cos^2(x) ?

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

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prove sin^2(x)(csc^2(x)-1)=cos^2(x)

prove

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

True

Solution steps

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

Manipulating left side

sin^{2} (x) (csc^{2} (x) - 1)

Express with sin, cos

(- 1 + csc^{2} (x)) sin^{2} (x)

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

(\frac{1}{sin ( x )})^{2}

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

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

Join

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

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

Convert element to fraction:

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

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

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

Multiply:

1 \cdot sin^{2} (x) = sin^{2} (x)

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin^{2} (x)

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

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

= 1 - sin^{2} (x)

Rewrite using trig identities

1 - sin^{2} (x)

Use the Pythagorean identity:

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

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

= cos^{2} (x)

= cos^{2} (x)

We showed that the two sides could take the same form

\Rightarrow True

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

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

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

cos (3 α) = 4 cos^{3} (α) - 3 cos (α)

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

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