Simplify tan(x)^4+2tan(x)^2+1

Problem

tan4(x)+2*tan2(x)+1

Solution

1. Identify the expression as a quadratic form by letting u=tan2(x) which transforms the expression into u2+2*u+1

2. Factor the perfect square trinomial using the identity a2+2*a*b+b2=(a+b)2

3. Substitute tan2(x) back into the factored form to get (tan2(x)+1)2

4. Apply the Pythagorean identity tan2(x)+1=sec2(x) to simplify the expression inside the parentheses.

5. Simplify the resulting power by applying the rule (am)n=a(m*n)

Final Answer

tan4(x)+2*tan2(x)+1=sec4(x)

Want more problems? Check here!