Verify the Identity tan(x)^2-sin(x)^2=tan(x)^2sin(x)^2

Problem

tan2(x)−sin2(x)=tan2(x)*sin2(x)

Solution

1. Rewrite the tangent function in terms of sine and cosine using the quotient identity tan(x)=sin(x)/cos(x)

tan2(x)−sin2(x)=sin2(x)/cos2(x)−sin2(x)

2. Factor out the common term sin2(x) from both terms in the expression.

sin2(x)*(1/cos2(x)−1)

3. Apply the reciprocal identity 1/cos2(x)=sec2(x) to simplify the expression inside the parentheses.

sin2(x)*(sec2(x)−1)

4. Substitute the Pythagorean identity sec2(x)−1=tan2(x) into the expression.

sin2(x)*tan2(x)

5. Rearrange the factors to match the right side of the original identity.

tan2(x)*sin2(x)

Final Answer

tan2(x)−sin2(x)=tan2(x)*sin2(x)

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