Question

If \((1 + \text{Cos} A)(1 - \text{Cos} A) = \frac{3}{4}\), then the value of Sec A is :

• A. 1
• B. 3
• C. 2
• D. 4

Hint:

Remember the algebraic identity $(a+b)(a-b) = a^2 - b^2$ and the fundamental trigonometric identity $\text{Sin}^2 A + \text{Cos}^2 A = 1$. These are crucial for simplifying trigonometric expressions. Also, $\text{Sec} A$ is the reciprocal of $\text{Cos} A$.

Answer 1

The Correct Option is C

Step 1: Simplify the given expression using algebraic identities.
The given equation is $(1 + \text{Cos} A)(1 - \text{Cos} A) = \frac{3}{4}$.
This expression is in the form of $(a+b)(a-b)$, which simplifies to $a^2 - b^2$.
Here, $a = 1$ and $b = \text{Cos} A$.
So, $(1)^2 - (\text{Cos} A)^2 = \frac{3}{4}$
$1 - \text{Cos}^2 A = \frac{3}{4}$
Step 2: Use the fundamental trigonometric identity. 
Recall the Pythagorean identity: $\text{Sin}^2 A + \text{Cos}^2 A = 1$.
From this identity, we can write $\text{Sin}^2 A = 1 - \text{Cos}^2 A$.
Substitute this into the simplified equation:
$\text{Sin}^2 A = \frac{3}{4}$ Step 3: Find the value of Sin A. 
Taking the square root of both sides:
$\text{Sin} A = \pm\sqrt{\frac{3}{4}} = \pm\frac{\sqrt{3}}{2}$
Step 4: Find the value of Cos A. 
We have $1 - \text{Cos}^2 A = \frac{3}{4}$.
Rearrange the equation to solve for $\text{Cos}^2 A$:
$\text{Cos}^2 A = 1 - \frac{3}{4}$
$\text{Cos}^2 A = \frac{4 - 3}{4} = \frac{1}{4}$
Taking the square root of both sides:
$\text{Cos} A = \pm\sqrt{\frac{1}{4}} = \pm\frac{1}{2}$
Step 5: Find the value of Sec A. 
Recall that $\text{Sec} A = \frac{1}{\text{Cos} A}$.
Using the value of $\text{Cos} A$:
$\text{Sec} A = \frac{1}{\pm\frac{1}{2}} = \pm 2$ Step 6: Choose the appropriate value from the options. 
The options provided are positive integers.

Therefore, we consider the positive value of \(\text{Sec} A = 2\)