MHTCET 26 Apr 2025 Shift 1 Paper

Concept:Use the cosine addition formula to simplify the given trigonometric equation into a linear form in $\cos \theta$ and $\sin \theta$.Explanation:Given: $\tan(\pi \cos \theta) = \cot(\pi \sin \theta)$.Rewrite $\tan$ and $\cot$ in terms of sine and cosine:$$\frac{\sin(\pi \cos \theta)}{\cos(\pi \cos \theta)} = \frac{\cos(\pi \sin \theta)}{\sin(\pi \sin \theta)}$$Cross-multiplying gives:$$\cos(\pi \cos \theta)\cos(\pi \sin \theta) - \sin(\pi \cos \theta)\sin(\pi \sin \theta) = 0$$Apply the identity $\cos(A+B) = \cos A \cos B - \sin A \sin B$:$$\cos(\pi \cos \theta + \pi \sin \theta) = 0$$Taking the principal value, $\pi(\cos \theta + \sin \theta) = \frac{\pi}{2}$.Hence:$$\cos \theta + \sin \theta = \frac{1}{2}$$Now express the left side using the sine addition formula:$$\cos \theta + \sin \theta = \sqrt{2}\left(\frac{1}{\sqrt{2}}\cos \theta + \frac{1}{\sqrt{2}}\sin \theta\right) = \sqrt{2}\sin\left(\frac{\pi}{4} + \theta\right)$$Substitute the earlier result:$$\sqrt{2}\sin\left(\frac{\pi}{4} + \theta\right) = \frac{1}{2}$$Therefore:$$\sin\left(\frac{\pi}{4} + \theta\right) = \frac{1}{2\sqrt{2}}$$Answer:$$\sin\left(\frac{\pi}{4} + \theta\right) = \frac{1}{2\sqrt{2}}$$