If $a = \sin \frac{\pi}{18} \sin \frac{5\pi}{18} \sin \frac{7\pi}{18}$ and $x$ is the solution of the equations $y = 2[x] + 2$ and $y = 3[x - 2]$,where $[x]$ denotes the greatest integer function of $x$,then $a$ is equal to:
- A$[x]$
- B$\frac{1}{[x]}$
- C$2[x]$
- D$[x]^2$
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Match the items of List-$I$ with those of the entries of List-$II$.
List-$I$ List-$II$ $(I)$ $\sin^2 5^{\circ} + \sin^2 10^{\circ} + \sin^2 15^{\circ} + \dots + \sin^2 90^{\circ}$ $(A)$ $0$ $(II)$ $\tan^2 5^{\circ} \cdot \tan^2 10^{\circ} \cdot \tan^2 15^{\circ} \dots \tan^2 85^{\circ}$ $(B)$ $\frac{19}{2}$ $(III)$ $\cos^2 5^{\circ} + \cos^2 10^{\circ} + \cos^2 15^{\circ} + \dots + \cos^2 180^{\circ}$ $(C)$ $18$ $(IV)$ $\cot 5^{\circ} + \cot 10^{\circ} + \cot 15^{\circ} + \dots + \cot 175^{\circ}$ $(D)$ $1$ $(E)$ $-1$
| List-$I$ | List-$II$ |
| $(I)$ $\sin^2 5^{\circ} + \sin^2 10^{\circ} + \sin^2 15^{\circ} + \dots + \sin^2 90^{\circ}$ | $(A)$ $0$ |
| $(II)$ $\tan^2 5^{\circ} \cdot \tan^2 10^{\circ} \cdot \tan^2 15^{\circ} \dots \tan^2 85^{\circ}$ | $(B)$ $\frac{19}{2}$ |
| $(III)$ $\cos^2 5^{\circ} + \cos^2 10^{\circ} + \cos^2 15^{\circ} + \dots + \cos^2 180^{\circ}$ | $(C)$ $18$ |
| $(IV)$ $\cot 5^{\circ} + \cot 10^{\circ} + \cot 15^{\circ} + \dots + \cot 175^{\circ}$ | $(D)$ $1$ |
| $(E)$ $-1$ |
If $\operatorname{cosec} \theta = \frac{p+q}{p-q}$,then $\cot \left(\frac{\pi}{4} + \frac{\theta}{2}\right)$ is equal to
$\operatorname{Tanh}^{-1}\left(\frac{1}{3}\right)+\operatorname{Coth}^{-1}(3)=$
DifficultAP EAMCET 2025
View SolutionThe value of $\cos ^2 76^{\circ}+\cos ^2 16^{\circ}-\cos 76^{\circ} \cos 16^{\circ}$ is equal to
If $\cos A = m \cos B,$ then
Medium
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