Verify the Identity csc(x)-csc(x)cos(x)^2=sin(x)

Problem

csc(x)−csc(x)*cos2(x)=sin(x)

Solution

1. Factor out the common term csc(x) from the left side of the equation.

csc(x)*(1−cos2(x))

2. Apply the Pythagorean identity sin2(x)+cos2(x)=1 which implies 1−cos2(x)=sin2(x)

csc(x)*sin2(x)

3. Substitute the reciprocal identity csc(x)=1/sin(x) into the expression.

1/sin(x)⋅sin2(x)

4. Simplify the expression by canceling one factor of sin(x)

sin(x)

Final Answer

csc(x)−csc(x)*cos2(x)=sin(x)

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