Is sin(x)csc^2(x)= 1/(sin(x)) ?

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

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prove sin(x)csc^2(x)= 1/(sin(x))

prove

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

True

Solution steps

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

Manipulating left side

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

Express with sin, cos

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

Use the basic trigonometric identity:

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

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

Simplify

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

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

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

Multiply fractions:

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

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

Multiply:

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

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

Cancel the common factor:

sin (x)

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

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

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

We showed that the two sides could take the same form

\Rightarrow True

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

(tan (θ + 1)) = sec (θ)

\frac{sin ( x ) csc ( x )}{cos ( x )} = sec (x)

tan (x) + sec (x) = 1

- sinh^{2} (x) + cosh^{2} (x) = 1

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