Question : Find the value of $\cos 47^\circ \sec 133^\circ + \sin 44^\circ \text{cosec}\; 136^\circ $.
Option 1: $\frac{1}{2}$
Option 2: $1$
Option 3: $0$
Option 4: $–1$
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Correct Answer: $0$
Solution :
We know that $\sec (90°+ \theta) = – \ \text{cosec}\; \theta$ and $\text{cosec} \;(90°+\theta) = \sec \theta$
$\cos 47^\circ \sec 133^\circ + \sin 44^\circ \ \text{cosec}\;136^\circ $
$=\cos 47^\circ \sec (90^\circ + 43^\circ) + \sin 44^\circ \ \text{cosec} \;(90^\circ + 46^\circ) $
$=-\cos 47^\circ \text{cosec}\;43^\circ + \sin 44^\circ \sec 46^\circ $
$=-\cos 47^\circ \text{cosec}\; (90^\circ - 47^\circ) + \sin 44^\circ \sec (90^\circ - 44^\circ) $
$=-\cos 47^\circ \sec \;47^\circ + \sin 44^\circ \ \text{cosec} \;44^\circ $
$=-1 +1=0$
Hence the correct answer is $0$.
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