prove sin(x)csc(x)=1

Solution

prove

sin (x) csc (x) = 1

True

Solution steps

sin (x) csc (x) = 1

Manipulating left side

sin (x) csc (x)

csc (x) sin (x) = 1

csc (x) sin (x)

Express with sin, cos

csc (x) sin (x)

Use the basic trigonometric identity:

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

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

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

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

Multiply fractions:

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

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

Cancel the common factor:

sin (x)

= 1

= 1

We showed that the two sides could take the same form

\Rightarrow True

sin (2 x) + cos (2 x) - 1 = 2 sin (x) (cos (x) - sin (x))

cos^{2} (θ) = 1 - sin^{2} (θ)

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

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

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