Simplify tan(x)^2-sin(x)^2

Problem

tan^2(x)−sin^2(x)

Solution

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

tan^2(x)−sin^2(x)=sin^2(x)/cos^2(x)−sin^2(x)

2. Factor out the common term sin^2(x) from both parts of the expression.

sin^2(x)*(1/cos^2(x)−1)

3. Apply the reciprocal identity 1/cos^2(x)=sec^2(x) to the term inside the parentheses.

sin^2(x)*(sec^2(x)−1)

4. Use the Pythagorean identity tan^2(x)+1=sec^2(x) which implies sec^2(x)−1=tan^2(x) to simplify the expression.

sin^2(x)*tan^2(x)

Final Answer

tan^2(x)−sin^2(x)=tan^2(x)*sin^2(x)

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