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

Problem

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

Solution

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

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

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

4. Apply the Pythagorean identity tan^2(x)+1=sec^2(x) to simplify the expression inside the parentheses.

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

Final Answer

tan^4(x)+2*tan^2(x)+1=sec^4(x)

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