Is sin(θ)csc(θ)=1 ?

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

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prove sin(θ)csc(θ)=1

prove

sin (θ) csc (θ) = 1

True

Solution steps

sin (θ) csc (θ) = 1

Manipulating left side

sin (θ) csc (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

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

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

\frac{tan ( θ )}{sec ( θ )} = sin (θ)

\frac{tan ( x )}{sin ( x )} = sec (x)

(1 - sin^{2} (θ)) sec^{2} (θ) = 1