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