Understanding the Trigonometric Expression

The question asks us to simplify the trigonometric expression \( \cot \left( \frac{A}{2} \right)-\tan \left( \frac{A}{2} \right) \). This involves using basic trigonometric ratios and identities to transform the expression into a simpler form, matching one of the given options. We will work with the half angles \( \frac{A}{2} \) and try to relate them to the angle \( A \).

Step-by-Step Solution for the Trigonometric Identity

Let's simplify the given expression \( \cot \left( \frac{A}{2} \right)-\tan \left( \frac{A}{2} \right) \). We can rewrite the cotangent and tangent in terms of sine and cosine:

The expression is: \( \cot \left( \frac{A}{2} \right)-\tan \left( \frac{A}{2} \right) \)

We know that \( \cot x = \frac{\cos x}{\sin x} \) and \( \tan x = \frac{\sin x}{\cos x} \). Applying this to our expression with \( x = \frac{A}{2} \), we get:

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

To combine these two fractions, we find a common denominator, which is \( \sin \left( \frac{A}{2} \right) \cos \left( \frac{A}{2} \right) \):

\( \frac{\cos \left( \frac{A}{2} \right) \cdot \cos \left( \frac{A}{2} \right) - \sin \left( \frac{A}{2} \right) \cdot \sin \left( \frac{A}{2} \right)}{\sin \left( \frac{A}{2} \right) \cos \left( \frac{A}{2} \right)} \)

This simplifies to:

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

Now, we can use double angle identities from trigonometry:

• We know that \( \cos(2x) = \cos^2 x - \sin^2 x \). If we let \( x = \frac{A}{2} \), then \( 2x = A \). So, \( \cos^2 \left( \frac{A}{2} \right) - \sin^2 \left( \frac{A}{2} \right) = \cos(A) \).
• We also know that \( \sin(2x) = 2 \sin x \cos x \). If we let \( x = \frac{A}{2} \), then \( 2x = A \). So, \( \sin(A) = 2 \sin \left( \frac{A}{2} \right) \cos \left( \frac{A}{2} \right) \). This means \( \sin \left( \frac{A}{2} \right) \cos \left( \frac{A}{2} \right) = \frac{1}{2} \sin(A) \).

Substitute these identities back into our expression:

\( \frac{\cos(A)}{\frac{1}{2} \sin(A)} \)

Simplify the fraction:

\( \frac{2 \cos(A)}{\sin(A)} \)

Finally, recall that \( \frac{\cos x}{\sin x} = \cot x \). So, \( \frac{\cos(A)}{\sin(A)} = \cot(A) \).

Therefore, the expression simplifies to \( 2 \cot(A) \).

Comparing this result with the given options:

• tan A
• cot A
• 2 tan A
• 2 cot A

Our simplified expression matches the fourth option, \( 2 \cot A \).

Revision Table: Key Trigonometric Identities

Understanding fundamental identities is crucial for simplifying trigonometric expressions like \( \cot \left( \frac{A}{2} \right)-\tan \left( \frac{A}{2} \right) \).

Additional Information on Half and Double Angle Formulas

The problem \( \cot \left( \frac{A}{2} \right)-\tan \left( \frac{A}{2} \right) \) highlights the relationship between angles \( A \) and \( A/2 \) using double-angle (or equivalently, half-angle) formulas. These formulas are derived from sum and difference identities.

• Double Angle Formulas:
  • \( \sin(2x) = 2 \sin x \cos x \)
  • \( \cos(2x) = \cos^2 x - \sin^2 x = 2 \cos^2 x - 1 = 1 - 2 \sin^2 x \)
  • \( \tan(2x) = \frac{2 \tan x}{1 - \tan^2 x} \)
• Half Angle Formulas (derived from double angle formulas):
  • \( \sin \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1 - \cos x}{2}} \)
  • \( \cos \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1 + \cos x}{2}} \)
  • \( \tan \left( \frac{x}{2} \right) = \pm \sqrt{\frac{1 - \cos x}{1 + \cos x}} = \frac{\sin x}{1 + \cos x} = \frac{1 - \cos x}{\sin x} \)

In our specific problem, we used the double angle formulas for sine and cosine in reverse to simplify the expression involving \( A/2 \) into terms of \( A \). The process involved converting the initial expression into a fraction involving \( \sin(A/2) \) and \( \cos(A/2) \), identifying the numerator as \( \cos A \) and the denominator as \( \frac{1}{2} \sin A \), ultimately leading to \( 2 \cot A \).