Understanding the Trigonometric Expression

The question asks us to find the number of values for the variable $\theta$ that satisfy the given equation:

$$ \sin^2 \theta + \cos^2 \theta - 1 = 0 $$

To solve this, we need to recall a fundamental identity in trigonometry.

Applying the Pythagorean Identity

One of the most basic and important trigonometric identities is the Pythagorean identity, which states:

$$ \sin^2 \theta + \cos^2 \theta = 1 $$

This identity holds true for any real value of $\theta$.

Simplifying the Equation

Now, let's substitute the value of $\sin^2 \theta + \cos^2 \theta$ from the identity into the given equation:

Original Equation: $ \sin^2 \theta + \cos^2 \theta - 1 = 0 $

Substitute using the identity: $ (1) - 1 = 0 $

Simplify the result: $ 0 = 0 $

Interpreting the Result

The equation simplifies to $0 = 0$. This is a true statement, regardless of the specific value chosen for $\theta$. This means the original expression is not an equation with a limited number of solutions, but rather a trigonometric identity that is true for all possible values of $\theta$.

• Since $\theta$ can be any real number (representing angles in degrees or radians), and the identity holds true for every single one of them, there isn't just one, two, or a finite number of solutions.
• The set of all possible values for $\theta$ is infinite.

Conclusion on Theta Values

Because the equation $ \sin^2 \theta + \cos^2 \theta - 1 = 0 $ simplifies to $0 = 0$, which is always true, the equation is satisfied by infinitely many values of $\theta$.