Class 11MathematicsTrigonometrical Ratios, Functions and IdentitiesMix Examples-Trigonometrical Ratios, Functions and Identities
MediumIf $A = \{x \in [0, 2\pi] : \tan x - \tan^2 x > 0\}$ and $B = \{x \in [0, 2\pi] : |\sin x| < \frac{1}{2}\}$,then $A \cap B =$
- A$\left(0, \frac{\pi}{6}\right) \cup \left(\pi, \frac{7\pi}{6}\right)$
- B$\left(0, \frac{\pi}{4}\right) \cup \left(\pi, \frac{7\pi}{6}\right)$
- C$\left(0, \frac{\pi}{6}\right) \cup \left(\frac{5\pi}{6}, \frac{7\pi}{6}\right)$
- D$\left(\frac{\pi}{6}, \frac{7\pi}{6}\right)$
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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 $2 \cos \theta + \sin \theta = 1$ $\left( \theta \neq \frac{\pi}{2} \right)$,then $7 \cos \theta + 6 \sin \theta$ is equal to
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MediumIIT 1977
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