Question : If (sin $\alpha$ + cosec $\alpha$)2 + (cos $\alpha$ + sec $\alpha$)2 = k + tan2 $\alpha$ + cot2 $\alpha$, then the value of k is:
Option 1: 1
Option 2: 7
Option 3: 3
Option 4: 5
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Correct Answer: 7
Solution :
(sin $\alpha$ + cosec $\alpha$)
2
+ (cos $\alpha$ + sec $\alpha$)
2
= k + tan
2
$\alpha$ + cot
2
$\alpha$
Taking the left-hand side of the expression, we get,
(sin $\alpha$ + cosec $\alpha$)
2
+ (cos $\alpha$ + sec $\alpha$)
2
= sin
2
$\alpha$ + cosec
2
$\alpha$ + 2sin $\alpha$cosec $\alpha$
+ cos
2
$\alpha$ + sec
2
$\alpha$ + 2cos $\alpha$sec $\alpha$
= (sin
2
$\alpha$ + cos
2
$\alpha$) + cosec
2
$\alpha$ + 2
+ sec
2
$\alpha$ + 2
We know that sin
2
$\alpha$ + cos
2
$\alpha$ = 1.
= 5 + cosec
2
$\alpha$ + sec
2
$\alpha$
= 5 + (1 + cot
2
$\alpha$) + (1 + tan
2
$\alpha$)
= 7 + tan
2
$\alpha$ + cot
2
$\alpha$
Comparing it with the right-hand side, we get, k = 7
Hence, the correct answer is 7.
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