Let \( S = \{\theta \in [0, 2\pi): \tan(\pi \cos \theta) + \tan(\pi \sin \theta) = 0\} \). Then \( \displaystyle \sum_{\theta \in S} \sin^{2}\!\left(\theta + \frac{\pi}{4}\right) \) is equal to __________.
In a triangle \( ABC \), if \( \cos A + 2 \cos B + \cos C = 2 \) and the lengths of the sides opposite to the angles \( A \) and \( C \) are \( 3 \) and \( 7 \) respectively, then \( \cos A - \cos C \) is equal to:
\( 96\cos\dfrac{\pi}{33}\cos\dfrac{2\pi}{33}\cos\dfrac{4\pi}{33}\cos\dfrac{8\pi}{33}\cos\dfrac{16\pi}{33} \) is equal to:
In the figure,\( \theta_{1} + \theta_{2} = \dfrac{\pi}{2} \) and \( \sqrt{3}(BE) = 4(AB) \). If the area of \( \triangle CAB \) is \( 2\sqrt{3} - 3 \) unit\( ^{2} \), when \( \theta_{2}/\theta_{1} \) is the largest, then the perimeter (in unit) of \( \triangle CED \) is equal to ______.
The value of \( 36(4\cos^{2}9^{\circ}-1)(4\cos^{2}27^{\circ}-1)(4\cos^{2}81^{\circ}-1)(4\cos^{2}243^{\circ}-1) \) is:
The number of elements in the set \( S = \left\{ \theta \in [0, 2\pi] : 3\cos^4 \theta - 5\cos^2 \theta - 2\sin^6 \theta + 2 = 0 \right\} \) is:
