Verify the Identity sec(x)^4-sec(x)^2=tan(x)^4+tan(x)^2

Problem

sec(x)−sec(x)=tan(x)+tan(x)

Solution

1. Factor the left side of the equation by taking out the greatest common factor, which is sec(x)

sec(x)−sec(x)=sec(x)*(sec(x)−1)

2. Apply the Pythagorean identity sec(x)=1+tan(x)

sec(x)−1=tan(x)

3. Substitute the identity into the factored expression from step 1.

sec(x)*(tan(x))

4. Substitute the identity sec(x)=1+tan(x) for the remaining secant term.

(1+tan(x))*(tan(x))

5. Distribute the tan(x) term across the parentheses to reach the final form.

tan(x)+tan(x)

Final Answer

sec(x)−sec(x)=tan(x)+tan(x)

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