Question

pr.route sec^2theta+cosec^2theta=tantheta+cot theta

Answer 1

Using the Pythagorean identities:

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

csc^2(x) = 1 + cot^2(x)

We may write:

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

Since tan(x)cot(x) = 1, we may then write:

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

Using the formula for the square of a binomials, we obtain:

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

Since the sum of two squares cannot be negative for real values, we may then write:

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

where x ≠ n(π/2)