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

Problem

cos2(x)*csc(x)−csc(x)=−sin(x)

Solution

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

csc(x)*(cos2(x)−1)

2. Apply the Pythagorean identity sin2(x)+cos2(x)=1 which can be rearranged to cos2(x)−1=−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) from the numerator and denominator.

−sin2(x)/sin(x)=−sin(x)

Final Answer

cos2(x)*csc(x)−csc(x)=−sin(x)

Want more problems? Check here!