Simplifying the Trigonometric Expression

The question asks us to simplify the following trigonometric expression:

To simplify this expression, we can use the technique of rationalizing the denominator. We multiply the numerator and the denominator inside the square root by the conjugate of the denominator, which is \(1+\cos \theta\).

Step 1: Rationalize the Denominator

Multiply the fraction inside the square root by \(\frac{1+\cos \theta}{1+\cos \theta}\):

This gives us:

Step 2: Apply Trigonometric Identities

Simplify the denominator using the difference of squares formula \((a-b)(a+b) = a^2 - b^2\):

Now, recall the fundamental Pythagorean trigonometric identity: \(\sin^2 \theta + \cos^2 \theta = 1\). Rearranging this gives us:

Substitute this back into the expression:

Step 3: Evaluate the Square Root

Now, take the square root of the numerator and the denominator:

This simplifies to:

Since \(\cos \theta\) is always greater than or equal to -1, \(1+\cos \theta\) is always non-negative. Therefore, \(|1+\cos \theta| = 1+\cos \theta\). Assuming \(\theta\) is in a range where \(\sin \theta\) is positive (like the first quadrant, \(0 < \theta < \pi\)), we have \(|\sin \theta| = \sin \theta\). So the expression becomes:

If \(\sin \theta\) is negative, the expression would be \(\frac{1+\cos \theta}{-\sin \theta}\). However, the standard simplification usually assumes the principal value or a range where the result matches the options provided.

Step 4: Separate Terms and Use Reciprocal Identities

Split the fraction into two terms:

Now, apply the reciprocal and quotient identities:

• \( \frac{1}{\sin \theta} = \text{cosec } \theta \)
• \( \frac{\cos \theta}{\sin \theta} = \cot \theta \)

Step 5: Final Simplified Expression

Substituting these back, we get the simplified expression:

This matches the fourth option.