Without using trigonometric tables, find the value of 4 tan 60 sec 30 + sin 31 sec 59 + cot 59 cot 31 / 8 sin 30 - tan ^2 45

Given expression:
(4 tan 60° sec 30° + sin 31° sec 59° + cot 59° cot 31°) / (8 sin 30° - tan² 45°)

Step 1: Evaluate trigonometric values
Using known exact values:

tan 60° = √3, sec 30° = 2/√3
sin 30° = 1/2, tan 45° = 1, so tan² 45° = 1
sec 59° = 1/cos 59° = 1/cos (90° - 31°) = 1/sin 31° = csc 31°
cot 59° = 1/tan 59° = tan (90° - 59°) = tan 31°
cot 31° = 1/tan 31°
Step 2: Evaluate each term
First term: 4 tan 60° sec 30° = 4 × (√3) × (2/√3)
= 4 × 2
= 8

Second term: sin 31° sec 59° = sin 31° × csc 31°
= 1

Third term: cot 59° cot 31° = tan 31° × (1/tan 31°)
= 1

Now, sum up the numerator: 8 + 1 + 1 = 10

Step 3: Evaluate denominator
8 sin 30° - tan² 45°
= 8 × (1/2) - 1
= 4 - 1
= 3

Step 4: Compute the final result
= 10 / 3
= 10/3

Thus, the final answer is 10/3.