If 8 cotθ = 7, then find the value of (1 + sinθ) / cosθ.

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Given: 8 cotθ = 7
=> cotθ = 7/8

Recall that cotθ = cosθ / sinθ
So, cosθ / sinθ = 7/8
=> cosθ = (7/8) sinθ

We need to find (1 + sinθ) / cosθ.

Using cosθ = (7/8) sinθ, substitute:
(1 + sinθ) / cosθ = (1 + sinθ) / ((7/8) sinθ) = (1 + sinθ) * (8 / 7 sinθ)

To simplify, express sinθ in terms of cotθ:
From cotθ = 7/8, tanθ = 8/7

Using identity: sin²θ + cos²θ = 1
Express sinθ and cosθ in terms of tanθ:

sinθ = tanθ / √(1 + tan²θ) = (8/7) / √(1 + (8/7)²) = (8/7) / √(1 + 64/49) = (8/7) / √(113/49) = (8/7) * (7/√113) = 8 / √113

Similarly, cosθ = 1 / √(1 + tan²θ) = 1 / √(1 + (8/7)²) = 1 / √(113/49) = 7 / √113

Now substitute sinθ and cosθ:

(1 + sinθ) / cosθ = (1 + 8/√113) / (7/√113) = ( (√113 / √113) + (8 / √113) ) / (7 / √113) = ( (√113 + 8) / √113 ) * (√113 / 7) = (√113 + 8) / 7

Therefore, the value is (8 + √113) / 7.

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