Evaluate Trigonometric Expression Given CosecA SecA Relation

The problem requires finding the value of a trigonometric expression given the relation $2\text{cosec}A = 3\text{sec}A$.

Simplify the Given Relation

Start by rewriting the given equation in terms of $\sin A$ and $\cos A$.

• The relation is $2\text{cosec}A = 3\text{sec}A$.
• Substitute $\text{cosec}A = \frac{1}{\sin A}$ and $\text{sec}A = \frac{1}{\cos A}$: $ \frac{2}{\sin A} = \frac{3}{\cos A} $
• Rearrange the terms to find $\tan A$: $ 2\cos A = 3\sin A $ $ \frac{\sin A}{\cos A} = \frac{2}{3} $ $ \tan A = \frac{2}{3} $

Simplify the Target Expression

The expression to evaluate is $\frac{(1 - \cos A)(1 + \cos A)}{(1 - \sin A)(1 + \sin A)}$. Use the difference of squares formula, $(a-b)(a+b) = a^2 - b^2$.

• Numerator: $(1 - \cos A)(1 + \cos A) = 1^2 - \cos^2 A = 1 - \cos^2 A$. Using the identity $\sin^2 A + \cos^2 A = 1$, we get $1 - \cos^2 A = \sin^2 A$.
• Denominator: $(1 - \sin A)(1 + \sin A) = 1^2 - \sin^2 A = 1 - \sin^2 A$. Using the identity $\sin^2 A + \cos^2 A = 1$, we get $1 - \sin^2 A = \cos^2 A$.
• The expression simplifies to: $ \frac{\sin^2 A}{\cos^2 A} $
• This is equal to $\tan^2 A$.

Calculate the Final Value

Substitute the value of $\tan A$ found earlier into the simplified expression $\tan^2 A$.

• We found $\tan A = \frac{2}{3}$.
• Therefore, $\tan^2 A = \left(\frac{2}{3}\right)^2$.
• $ \tan^2 A = \frac{4}{9} $

The value of the expression $\frac{(1 - \cos A)(1 + \cos A)}{(1 - \sin A)(1 + \sin A)}$ is $\frac{4}{9}$.