To verify the claim, we can start by manipulating the expression on the left side of the equation.
Given: (1 + tan(theta))^2 = 1/cos(theta)
Using the identity tan^2(theta) + 1 = sec^2(theta), we can rewrite the left side of the equation:
(1 + tan(theta))^2 = (1 + tan(theta))^2 * (1 + tan(theta))^0
= (1 + tan(theta))^2 * (sec^2(theta))^0
= (1 + tan(theta))^2 * (1/cos^2(theta))^0
= (1 + tan(theta))^2 * 1
= (1 + tan(theta))^2
Now, let's simplify the right side of the equation:
1/cos(theta) = cos^(-1)(theta)
= sec(theta)
So our equation becomes:
(1 + tan(theta))^2 = sec(theta)
Next, we'll use the identity sec(theta) = 1/cos(theta):
(1 + tan(theta))^2 = 1/cos(theta)
Now, we can square root both sides of the equation:
1 + tan(theta) = ±sqrt(1/cos(theta))
Squaring both sides again:
(1 + tan(theta))^2 = 1/cos(theta)
We've reached the original equation, so by substituting the right side back in, we have:
1/cos(theta) = 1/cos(theta)
Therefore, the claim is true for all theta satisfying -π/2 < theta < π/2.