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

Problem

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

Solution

1. Expand the squared binomial using the identity (a+b)2=a2+2*a*b+b2

(tan(x)+cot(x))2=tan2(x)+2*tan(x)*cot(x)+cot2(x)

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

tan2(x)+2*(1)+cot2(x)

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

tan2(x)+1+cot2(x)+1

4. Apply the Pythagorean identities tan2(x)+1=sec2(x) and cot2(x)+1=csc2(x)

sec2(x)+csc2(x)

Final Answer

(tan(x)+cot(x))2=sec2(x)+csc2(x)

Want more problems? Check here!