If (7sinθ + 4cosθ) / (9sinθ - 2cosθ) = 5/4, then find the value of (tan²θ + 5) / (tan²θ - 5).

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Given: (7sinθ + 4cosθ) / (9sinθ - 2cosθ) = 5/4

Let tanθ = t.

Rewrite sinθ and cosθ in terms of t:

sinθ = t / √(1 + t²), cosθ = 1 / √(1 + t²)

Substitute into the equation:

(7 * (t / √(1 + t²)) + 4 * (1 / √(1 + t²))) / (9 * (t / √(1 + t²)) - 2 * (1 / √(1 + t²))) = 5/4

Multiply numerator and denominator by √(1 + t²):

(7t + 4) / (9t - 2) = 5/4

Cross multiply:

4(7t + 4) = 5(9t - 2)

28t + 16 = 45t - 10

Bring terms to one side:

28t - 45t = -10 - 16

-17t = -26

t = 26/17

Now find (tan²θ + 5) / (tan²θ - 5):

Calculate tan²θ = (26/17)² = 676 / 289

Numerator:
676/289 + 5 = (676 + 5*289) / 289 = (676 + 1445) / 289 = 2121 / 289

Denominator:
676/289 - 5 = (676 - 1445) / 289 = (-769) / 289

Therefore,

(tan²θ + 5) / (tan²θ - 5) = (2121 / 289) / (-769 / 289) = 2121 / 289 * 289 / (-769) = -2121 / 769

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