Understanding the Trigonometric Expression cot A + cosec A

The question asks us to find an equivalent expression for the sum of cotangent A and cosecant A, represented as \(\cot A + \operatorname{cosec} A\). We need to simplify this expression and match it with one of the given options, which involve half-angle trigonometric functions.

Step-by-Step Simplification

We can start by expressing \(\cot A\) and \(\operatorname{cosec} A\) in terms of sine and cosine functions. The fundamental identities are:

• \(\cot A = \frac{\cos A}{\sin A}\)
• \(\operatorname{cosec} A = \frac{1}{\sin A}\)

Now, substitute these into the given expression:

\(\cot A + \operatorname{cosec} A = \frac{\cos A}{\sin A} + \frac{1}{\sin A}\)

Since both terms have a common denominator \(\sin A\), we can combine them:

\(\cot A + \operatorname{cosec} A = \frac{\cos A + 1}{\sin A} = \frac{1 + \cos A}{\sin A}\)

To relate this expression to the half-angle options, we can use trigonometric identities for \(1 + \cos A\) and \(\sin A\) in terms of \(\frac{A}{2}\). The relevant half-angle or double-angle identities (derived from \(\cos(2x) = 2\cos^2 x - 1\) and \(\sin(2x) = 2\sin x \cos x\)) are:

• \(\cos(2x) = 2\cos^2 x - 1 \implies 1 + \cos(2x) = 2\cos^2 x\). Replacing \(2x\) with \(A\), we get \(1 + \cos A = 2\cos^2 \left( \frac{A}{2} \right)\).
• \(\sin(2x) = 2\sin x \cos x\). Replacing \(2x\) with \(A\), we get \(\sin A = 2\sin \left( \frac{A}{2} \right) \cos \left( \frac{A}{2} \right)\).

Substitute these identities into the simplified expression \(\frac{1 + \cos A}{\sin A}\):

\(\frac{1 + \cos A}{\sin A} = \frac{2\cos^2 \left( \frac{A}{2} \right)}{2\sin \left( \frac{A}{2} \right) \cos \left( \frac{A}{2} \right)}\)

Now, we can cancel out the common terms from the numerator and the denominator. The constant 2 cancels out, and one term of \(\cos \left( \frac{A}{2} \right)\) cancels out:

\(\frac{2\cos^2 \left( \frac{A}{2} \right)}{2\sin \left( \frac{A}{2} \right) \cos \left( \frac{A}{2} \right)} = \frac{\cos \left( \frac{A}{2} \right)}{\sin \left( \frac{A}{2} \right)}\)

Finally, we recognize that the ratio of cosine to sine is the cotangent function:

\(\frac{\cos \left( \frac{A}{2} \right)}{\sin \left( \frac{A}{2} \right)} = \cot \left( \frac{A}{2} \right)\)

Thus, \(\cot A + \operatorname{cosec} A\) is equal to \(\cot \left( \frac{A}{2} \right)\).

Comparing with Options

Let's compare our simplified expression with the given options:

• Option 1: \(\tan \left( \frac{A}{2} \right)\)
• Option 2: \(\cot \left( \frac{A}{2} \right)\)
• Option 3: \(2\tan \left( \frac{A}{2} \right)\)
• Option 4: \(2\cot \left( \frac{A}{2} \right)\)

Our derived expression \(\cot \left( \frac{A}{2} \right)\) matches Option 2.

Revision Table: Key Trigonometric Identities

Additional Information on Trigonometric Simplification

Simplifying trigonometric expressions often involves converting functions to their sine and cosine forms, finding common denominators, and applying fundamental identities like Pythagorean identities, sum/difference formulas, double angle formulas, or half-angle formulas. Recognizing patterns and knowing these identities is crucial for simplifying complex expressions and solving trigonometric equations. The expression \(\cot A + \operatorname{cosec} A\) is a common one that simplifies neatly using half-angle identities. Another related expression is \(\tan A + \sec A\), which simplifies to \(\tan \left( \frac{\pi}{4} + \frac{A}{2} \right)\).