Is csc(θ)*cos(θ)=cot(θ) ?

The answer to whether csc(θ)*cos(θ)=cot(θ) is True

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prove csc(θ)*cos(θ)=cot(θ)

prove

csc (θ) \cdot cos (θ) = cot (θ)

True

Solution steps

csc (θ) cos (θ) = cot (θ)

Manipulating left side

csc (θ) cos (θ)

Express with sin, cos

csc (θ) cos (θ)

Use the basic trigonometric identity:

csc (θ) = \frac{1}{sin ( θ )}

= \frac{1}{sin ( θ )} cos (θ)

Simplify

\frac{1}{sin ( θ )} cos (θ) : \frac{cos ( θ )}{sin ( θ )}

\frac{1}{sin ( θ )} cos (θ)

Multiply fractions:

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

= \frac{1 cos ( θ )}{sin ( θ )}

Multiply:

1 \cdot cos (θ) = cos (θ)

= \frac{cos ( θ )}{sin ( θ )}

= \frac{cos ( θ )}{sin ( θ )}

= \frac{cos ( θ )}{sin ( θ )}

Use the basic trigonometric identity:

\frac{cos ( θ )}{sin ( θ )} = cot (θ)

= cot (θ)

We showed that the two sides could take the same form

\Rightarrow True

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

2 cos (4 θ) + 5 = 16 sin^{2} (θ)

(1 + cos (x)) (csc (x) - cot (x)) = sin (x)