Final answer:
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).