Prove that (csc theta+cot theta)(1-cos theta)=sin theta

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

cot(x) = cos(x) / sin(x)
csc(x) = 1 / sin(x)

Step-by-step solution:

• Substitute values into equation

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

• Simplify

-> ( (1 + cos(x) ) / sin(x) )(1 - cos(x) ) = sin(x)

-> [ ( 1 + cos(x) ) ( 1 - cos(x) ) ] / sin(x) = sin(x)

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

Substitute it into equation,

-> sin(x)² / sin(x) = sin(x)

-> sin(x)² = sin(x) • sin(x)

-> sin(x)² = sin(x)²

-> sin(x)² - sin(x)² = 0

-> 0 = 0

Hence,
( csc(x) + cot(x) )(1 - cos(x) ) = sin(x) --> True since both sides (0=0) are equal