Q.T \( \sin ^{5} \theta+\cos ^{2} \theta=\frac{1}{2^{6}}[\sin 7 \theta-3 \sin 5 \theta+\sin 3 \theta+5 \sin \theta] \)

To solve the equation \( \sin ^{5} \theta+\cos ^{2} \theta=\frac{1}{2^{6}}[\sin 7 \theta-3 \sin 5 \theta+\sin 3 \theta+5 \sin \theta] \), we can utilize some trigonometric identities and transformations. Firstly, we know that \( \cos^2 \theta = 1 - \sin^2 \theta \). So, we can express the left side as: \[ \sin^5 \theta + (1 - \sin^2 \theta) = \sin^5 \theta - \sin^2 \theta + 1 \] Now, let’s tackle the right side. The expression inside the brackets can be simplified using the angle addition formulas or by recognizing the patterns with sin terms. Each sine function can be related back to speeds of oscillation. Next, we can use numerical or graphical methods to investigate values of \(\theta\) to gain insights into possible solutions. It’s often helpful to think about specific angles, like \( \theta = 0, \frac{\pi}{4}, \frac{\pi}{2} \) to test their validity. To analyze the behavior of the equation more effectively, you can take numerical approximations or plots to identify intersection points. Alternatively, try substituting \( \sin \theta = x \) to convert the trigonometric equation into a polynomial equation for easier handling. This equation would generally require particular attention on the angle parameters while ensuring to check if there are periodic solutions, especially since the sine functions are periodic.