Question : The value of $\frac{\sin 25^{\circ}\cos 65^{\circ}+\cos 25^{\circ}\sin 65^{\circ}}{\tan ^{2}70^{\circ}-cosec^{2}20^{\circ}}$ is:
Option 1: –1
Option 2: 0
Option 3: 1
Option 4: 2
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Correct Answer: –1
Solution :
$\frac{\sin 25^{\circ}\cos 65^{\circ}+\cos 25^{\circ}\sin 65^{\circ}}{\tan ^{2}70^{\circ}-\operatorname{cosec}^{2}20^{\circ}}$
$=\frac{\sin 25^{\circ}\sin 25^{\circ}+\cos 25^{\circ}\cos 25^{\circ}}{\tan ^{2}70^{\circ}-\operatorname{cosec}^{2}20^{\circ}}$
$= \frac{\sin^2 25^{\circ}+\cos^225^{\circ}}{\tan^{2}70^{\circ}-\sec^{2}70^{\circ}}$
$= \frac{1}{-(\sec^ {2}70^{\circ}-\tan^ {2}70^{\circ})}$
$= -1$
Hence, the correct answer is –1.
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