Verify the Identity (tan(x)*sin(x))/(sec(x)^2-1)=cos(x)

Problem

(tan(x)*sin(x))/(sec2(x)−1)=cos(x)

Solution

1. Identify the Pythagorean identity for the denominator. Recall that 1+tan2(x)=sec2(x) which implies sec2(x)−1=tan2(x)

2. Substitute the identity into the denominator of the left side of the equation.

(tan(x)*sin(x))/tan2(x)

3. Simplify the fraction by canceling one factor of tan(x) from the numerator and the denominator.

sin(x)/tan(x)

4. Rewrite the tangent function using the quotient identity tan(x)=sin(x)/cos(x)

sin(x)/sin(x)/cos(x)

5. Multiply by the reciprocal of the denominator to simplify the complex fraction.

sin(x)⋅cos(x)/sin(x)

6. Cancel the common factor of sin(x) to reach the final expression.

cos(x)

Final Answer

(tan(x)*sin(x))/(sec2(x)−1)=cos(x)

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