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.