If tan θ = 5/8, then find the value of (1 + cos θ)(1 - cos θ) / (1 + sin θ)(1 - sin θ).

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Given tan θ = 5/8.

Step 1: Express sin θ and cos θ in terms of tan θ.

We know tan θ = sin θ / cos θ = 5/8.

Let cos θ = x.
Then sin θ = (5/8) * x.

Using the Pythagorean identity:

sin² θ + cos² θ = 1
=> ((5/8) * x)² + x² = 1
=> (25/64) x² + x² = 1
=> (25/64 + 1) x² = 1
=> (25/64 + 64/64) x² = 1
=> (89/64) x² = 1
=> x² = 64/89
=> cos θ = ±8/√89

Since tan θ = positive (5/8), θ is in the first quadrant where sin and cos are positive.
So, cos θ = 8/√89.

Then sin θ = (5/8) * (8/√89) = 5/√89.

Step 2: Calculate numerator and denominator separately.

Numerator:
(1 + cos θ)(1 - cos θ) = 1 - cos² θ = 1 - (8/√89)² = 1 - 64/89 = (89 - 64)/89 = 25/89.

Denominator:
(1 + sin θ)(1 - sin θ) = 1 - sin² θ = 1 - (5/√89)² = 1 - 25/89 = (89 - 25)/89 = 64/89.

Step 3: Calculate the expression:

(1 + cos θ)(1 - cos θ) / (1 + sin θ)(1 - sin θ) = (25/89) / (64/89) = 25/89 * 89/64 = 25/64.

Wait, this contradicts the earlier conclusion. Let's re-check.

Re-check Step 3:

(25/89) / (64/89) = (25/89) * (89/64) = 25/64.

So the value is 25/64.

Step 4: Final answer is 25/64.

But the options show 25/64 as option 4.

Hence, the correct answer is 25/64.

Note: The initial marking of option 3 as correct was a mistake. The correct answer is option 4.

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