Is csc(θ)sin(θ)=1 ?

The answer to whether csc(θ)sin(θ)=1 is True

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prove csc(θ)sin(θ)=1

prove

csc (θ) sin (θ) = 1

True

Solution steps

csc (θ) sin (θ) = 1

Manipulating left side

csc (θ) sin (θ)

csc (θ) sin (θ) = 1

csc (θ) sin (θ)

Express with sin, cos

csc (θ) sin (θ)

Use the basic trigonometric identity:

csc (θ) = \frac{1}{sin ( θ )}

= \frac{1}{sin ( θ )} sin (θ)

\frac{1}{sin ( θ )} sin (θ) = 1

\frac{1}{sin ( θ )} sin (θ)

Multiply fractions:

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

= \frac{1 sin ( θ )}{sin ( θ )}

Cancel the common factor:

sin (θ)

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

\frac{cos ( x ) - cos ( 9 x )}{sin ( x ) + sin ( 9 x )} = tan (4 x)

\frac{2 - sec ^{2} ( x )}{sec ^{2} ( x ) + 2 tan ( x )} = \frac{cos ( x ) - sin ( x )}{cos ( x ) + sin ( x )}

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

cos^{2} (9 x) - sin^{2} (9 x) = cos (18 x)

1 + cot^{2} (3 0^{\circ}) = csc^{2} (3 0^{\circ})