The equation in the image is:

$\sec^2 \theta + 3 \csc^2 \theta = 8$

Let’s solve for $\theta$.

Step 1: Recall the basic trigonometric identities.

• $\sec^2 \theta = 1 + \tan^2 \theta$
• $\csc^2 \theta = 1 + \cot^2 \theta$

These identities can be substituted into the equation to simplify.

Step 2: Substituting the identities

$(1 + \tan^2 \theta) + 3(1 + \cot^2 \theta) = 8$

Step 3: Expand the equation

$1 + \tan^2 \theta + 3 + 3 \cot^2 \theta = 8$ $\tan^2 \theta + 3 \cot^2 \theta + 4 = 8$

Step 4: Simplify the equation

$\tan^2 \theta + 3 \cot^2 \theta = 4$

Step 5: Express $\cot^2 \theta$ in terms of $\tan^2 \theta$

We know that: $\cot^2 \theta = \frac{1}{\tan^2 \theta}$

So, the equation becomes: $\tan^2 \theta + \frac{3}{\tan^2 \theta} = 4$

Step 6: Multiply the equation by $\tan^2 \theta$ to eliminate the denominator

Let $x = \tan^2 \theta$. Then, we get: $x^2 + 3 = 4x$ $x^2 - 4x + 3 = 0$

Step 7: Solve the quadratic equation

$(x - 3)(x - 1) = 0$ Thus, $x = 3$ or $x = 1$.

Step 8: Solve for $\theta$

Since $x = \tan^2 \theta$, we have two cases:

• If $\tan^2 \theta = 3$, then $\tan \theta = \sqrt{3}$, which gives $\theta = 60^\circ$ or $\theta = 240^\circ$.
• If $\tan^2 \theta = 1$, then $\tan \theta = 1$, which gives $\theta = 45^\circ$ or $\theta = 225^\circ$.

Thus, the possible values of $\theta$ are $45^\circ$, $60^\circ$, $225^\circ$, or $240^\circ$.

Would you like more details or clarifications?

Here are 5 related questions:

1. What is the general solution for $\tan \theta = 1$?
2. How would the equation change if the constant on the right-hand side were different?
3. Can you solve this equation graphically using trigonometric curves?
4. What other identities might help simplify equations involving $\sec^2 \theta$ and $\csc^2 \theta$?
5. How can you verify the solutions of a trigonometric equation using a calculator?

Tip: Always remember to check if your solutions lie within the required domain when solving trigonometric equations.