We are given the equation:

$\sec^2\theta + \tan^2\theta = 7$

Step 1: Using a trigonometric identity

We know from trigonometric identities that:

$\sec^2\theta = 1 + \tan^2\theta$

Substitute $\sec^2\theta$ into the original equation:

$(1 + \tan^2\theta) + \tan^2\theta = 7$

Simplifying:

$1 + 2\tan^2\theta = 7$

$2\tan^2\theta = 6$

$\tan^2\theta = 3$

Step 2: Solving for $\tan\theta$

Taking the square root of both sides:

$\tan\theta = \pm\sqrt{3}$

Step 3: Using another identity to find $\sin\theta$

We know the identity:

$\sin^2\theta = \frac{\tan^2\theta}{1 + \tan^2\theta}$

Substituting $\tan^2\theta = 3$:

$\sin^2\theta = \frac{3}{1 + 3} = \frac{3}{4}$

Taking the square root:

$\sin\theta = \pm\frac{\sqrt{3}}{2}$

Therefore, the value of $\sin\theta$ is either $\frac{\sqrt{3}}{2}$ or $-\frac{\sqrt{3}}{2}$, depending on the quadrant of $\theta$.

Would you like more details on how to determine the quadrant?

Here are 5 related questions:

1. What are the values of trigonometric functions in different quadrants?
2. How is the Pythagorean identity used in solving trigonometric equations?
3. What is the relationship between $\sec\theta$ and $\cos\theta$?
4. How do you determine the sign of $\sin\theta$ based on $\tan\theta$?
5. How can you apply the tangent identity in solving other trigonometric problems?

Tip: Always consider the quadrant to determine the correct sign of trigonometric values.