If sec$^2$θ + tan$^2$θ = 3, then the angle θ is equal to?

Analysis

We are given a trigonometric equation involving $\sec^2\theta$ and $\tan^2\theta$. To solve for $\theta$, we need to express the entire equation in terms of a single trigonometric function using standard identities.

Concept

The fundamental trigonometric identity relating secant and tangent is:

By substituting this identity into the given equation, we can reduce the problem to a simple algebraic equation.

Step-by-Step Solution

Step 1: Substitute the identity Given: $\sec^2\theta + \tan^2\theta = 3$ Since $\sec^2\theta = 1 + \tan^2\theta$, substitute this into the equation: $(1 + \tan^2\theta) + \tan^2\theta = 3$

Step 2: Simplify the equation Combine the like terms ($\tan^2\theta$): $1 + 2\tan^2\theta = 3$

Step 3: Solve for $\tan^2\theta$ Subtract 1 from both sides: $2\tan^2\theta = 3 - 1$ $2\tan^2\theta = 2$

Divide by 2: $\tan^2\theta = 1$

Step 4: Find $\theta$ Take the square root of both sides: $\tan\theta = \sqrt{1}$ $\tan\theta = 1$

Now, find the angle whose tangent is 1: $\theta = \tan^{-1}(1)$ $\theta = 45^\circ$

Conclusion

Comparing our result with the provided options, we find that $45^\circ$ corresponds to option C.

Exam Tip

Always remember the "Pythagorean Identities" for trigonometry:

1. $\sin^2\theta + \cos^2\theta = 1$
2. $1 + \tan^2\theta = \sec^2\theta$
3. $1 + \cot^2\theta = \csc^2\theta$

If you see a mix of squared trig functions in an exam, your first step should always be to use these identities to convert them into the same function!