Question

If \( \cot A = \frac{b}{a} \), then prove that \( \frac{b \sec A}{a \operatorname{cosec} A} = 1 \).

Hint:

A common and effective strategy for proving trigonometric identities is to express all trigonometric ratios in terms of sine and cosine. This often simplifies the expression and makes the path to the solution clearer.

Answer 1

Step 1: Understanding the Concept:
We need to prove the given trigonometric identity by simplifying the Left Hand Side (LHS) of the equation and showing that it equals the Right Hand Side (RHS), which is 1. We will use fundamental trigonometric identities and the given value of \( \cot A \).
Step 2: Key Formula or Approach:
We will use the following reciprocal and quotient identities:
1. \( \sec A = \frac{1}{\cos A} \)
2. \( \operatorname{cosec} A = \frac{1}{\sin A} \)
3. \( \tan A = \frac{\sin A}{\cos A} \)
4. \( \tan A = \frac{1}{\cot A} \)
Step 3: Detailed Explanation:
Let's start with the Left Hand Side (LHS) of the equation:
\[ \text{LHS} = \frac{b \sec A}{a \operatorname{cosec} A} \] Substitute the reciprocal identities for \( \sec A \) and \( \operatorname{cosec} A \):
\[ \text{LHS} = \frac{b \left( \frac{1}{\cos A} \right)}{a \left( \frac{1}{\sin A} \right)} \] Simplify the complex fraction:
\[ \text{LHS} = \frac{b}{a} \times \frac{\sin A}{\cos A} \] Using the quotient identity \( \tan A = \frac{\sin A}{\cos A} \):
\[ \text{LHS} = \frac{b}{a} \tan A \] We are given that \( \cot A = \frac{b}{a} \). We know that \( \tan A = \frac{1}{\cot A} \).
Therefore, \( \tan A = \frac{1}{\frac{b}{a}} = \frac{a}{b} \).
Now, substitute this value of \( \tan A \) back into the LHS expression:
\[ \text{LHS} = \frac{b}{a} \times \left( \frac{a}{b} \right) \] \[ \text{LHS} = 1 \] Since LHS = RHS, the identity is proved.
\[ \text{Hence Proved.} \]