Explanation of the given expression:

The given expression is:

√(1 - cosθ) / (1 - cosθ) √(1 - cosθ) / cosθ

To simplify this expression, we can break it down into its individual components and then evaluate each part separately.

1. Simplifying the numerator:

The numerator of the expression is √(1 - cosθ). We can simplify this by rationalizing the denominator.

√(1 - cosθ) * √(1 + cosθ) / √(1 + cosθ)

Using the property of square roots, we can multiply the numerator and denominator by the conjugate of the denominator to eliminate the square root.

(√(1 - cosθ) * √(1 + cosθ)) / (√(1 + cosθ) * √(1 + cosθ))

Simplifying further:

√((1 - cosθ)(1 + cosθ)) / (1 + cosθ)

Using the property (a - b)(a + b) = a^2 - b^2, we can simplify the numerator:

√(1 - cos^2θ) / (1 + cosθ)

2. Simplifying the denominator:

The denominator of the expression is (1 - cosθ) √(1 - cosθ) / cosθ. We can simplify this using the distributive property.

(1 - cosθ) * √(1 - cosθ) / cosθ

Expanding the numerator:

√(1 - cosθ - cos^2θ) / cosθ

Using the property of square roots, we can simplify the numerator:

√((1 - cosθ)(1 + cosθ)) / cosθ

Again, using the property (a - b)(a + b) = a^2 - b^2, we can simplify the numerator:

√(1 - cos^2θ) / cosθ

3. Combining the simplified numerator and denominator:

Now that we have simplified the numerator and denominator separately, we can combine them:

(√(1 - cos^2θ) / (1 + cosθ)) / (√(1 - cos^2θ) / cosθ)

To divide fractions, we can multiply by the reciprocal of the second fraction:

(√(1 - cos^2θ) / (1 + cosθ)) * (cosθ / √(1 - cos^2θ))

Simplifying further:

cosθ / (1 + cosθ)

Final Answer:

The simplified expression of the given expression is cosθ / (1 + cosθ).