Question

Why is sec^2(x)+1/sec^2(x)= sin^2x+2cos^2x

Answer 1

The equation sec^2(x) + 1/sec^2(x) = sin^2x + 2cos^2x is proved by using the identity sec^2(x) = 1 + tan^2(x) and simplifying the equation.

Step-by-step explanation:

The given equation is:

sec2(x) + 1/sec2(x) = sin2(x) + 2cos2(x)

To prove this equation, we can start with the identity: sec2(x) = 1 + tan2(x)

Substituting this identity into the given equation, we have:

(1 + tan2(x)) + 1/(1 + tan2(x)) = sin2(x) + 2cos2(x)

Simplifying this equation, we get:

1 + 1 + tan2(x)/(1 + tan2(x)) = sin2(x) + 2cos2(x)

Cancelling out the common terms, we have:

2 = sin2(x) + 2cos2(x)

Hence, we have proved that sec2(x) + 1/sec2(x) = sin2(x) + 2cos2(x).