The number of solutions of the pair of equations $$ \begin{aligned} &2 \sin ^{2} \theta-\cos 2 \theta=0 \\ &2 \cos ^{2} \theta-3 \sin \theta=0 \end{aligned} $$ in the interval \([0,2 \pi]\) is (A) zero (B) one (C) two (D) four

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables involved. These identities are very useful when solving trigonometric equations because they allow us to manipulate and simplify complex expressions. For example, one commonly used identity is \( \text{cos}(2\theta) = 1 - 2\sin^2(\theta) \), which expresses the cosine of a double angle in terms of the sine of the original angle.

There are many such identities, distinguishing between Pythagorean identities, angle sum and difference identities, double angle identities, and others. Each serves a unique purpose during the problem-solving process. In our exercise, using a double angle identity allowed the equation to be rephrased entirely in terms of \( \text{sin}(\theta) \), streamlining the pathway to our solution.

Solving trigonometric equations involves finding the values of the angles that satisfy the equation within a given interval. This can be a complex task as trigonometric functions are periodic, meaning they repeat values at regular intervals. A structured approach is critical.

The process generally includes: isolating the trigonometric function, using identities to simplify, expressing the solutions in terms of the appropriate trigonometric ratios, and ensuring the solutions fall within the specified interval. Remember that when we take the square root of both sides of an equation like \( \sin^2(\theta) = 1/4 \), we consider both positive and negative roots because the sine function can have both positive and negative values.

The interval within which we study a trigonometric function is essential as it dictates the number and range of possible solutions. Since trigonometric functions are periodic, they can have an infinite number of solutions across all real numbers. However, by limiting the interval, we confine these solutions to a manageable set.

Common intervals include \( [0, 2\pi] \) or \( [-\pi, \pi] \), corresponding to one full rotation around a unit circle. In the given exercise, we're considering the interval \( [0, 2\pi] \), which means we're looking for angles from 0 up to but not including a full rotation. When \( \sin(\theta) = 1/2 \), for example, we find that the angles \( \pi/6 \) and \( 5\pi/6 \) fall within our interval, making them valid solutions.

Cross-referencing is an important verification step when solving multiple trigonometric equations simultaneously. After finding potential solutions for each equation independently, we cross-reference them to determine which solutions satisfy all of the equations involved.

For instance, if certain angles satisfy one equation but not the other, we cannot consider them as valid solutions to the system of equations. In our original problem, after finding solutions for each equation independently, we verify that only \( \pi/6 \) and \( 5\pi/6 \) satisfy both equations within the given interval. We discard the other angles because they don't satisfy both equations, ensuring accurate and complete solutions.