Simplify (tan(x)+cot(x))^2

Problem

(tan(x)+cot(x))^2

Solution

1. Expand the squared binomial using the identity (a+b)^2=a^2+2*a*b+b^2

(tan(x)+cot(x))^2=tan^2(x)+2*tan(x)*cot(x)+cot^2(x)

2. Simplify the middle term using the reciprocal identity cot(x)=1/tan(x) which implies tan(x)*cot(x)=1

tan^2(x)+2*(1)+cot^2(x)

3. Rearrange the terms to group them for Pythagorean identities.

tan^2(x)+1+cot^2(x)+1

4. Apply the Pythagorean identities tan^2(x)+1=sec^2(x) and cot^2(x)+1=csc^2(x)

sec^2(x)+csc^2(x)

Final Answer

(tan(x)+cot(x))^2=sec^2(x)+csc^2(x)

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