If `secx+tanx=a`, then find the value of `sinx`.

Options

A. `(a^(2)-1)/(a^(2)+1)`
B. `(a^(2)+1)/(a^(2)-1)`
C. `(a-1)/(a^(2)+1)`
D. `(a^(2)-1)/(a+1)`

Answer

To find the value of \( \sin x \) given that \( \sec x + \tan x = a \), we can follow these steps:

### Step 1: Use the identity for \( \sec x + \tan x \)
We know that:
\[ \sec x + \tan x = a \]
We can also use the identity:
\[ \sec^2 x - \tan^2 x = 1 \]
This can be factored as:
\[ (\sec x + \tan x)(\sec x - \tan x) = 1 \]

### Step 2: Substitute \( \sec x + \tan x \)
Substituting \( \sec x + \tan x = a \) into the identity gives us:
\[ a(\sec x - \tan x) = 1 \]
From this, we can solve for \( \sec x - \tan x \):
\[ \sec x - \tan x = \frac{1}{a} \]

### Step 3: Set up equations for \( \sec x \) and \( \tan x \)
Now we have two equations:
1. \( \sec x + \tan x = a \)
2. \( \sec x - \tan x = \frac{1}{a} \)

Let’s denote \( \sec x = S \) and \( \tan x = T \). We can rewrite our equations as:
1. \( S + T = a \)
2. \( S - T = \frac{1}{a} \)

### Step 4: Solve for \( S \) and \( T \)
Adding these two equations:
\[ (S + T) + (S - T) = a + \frac{1}{a} \]
This simplifies to:
\[ 2S = a + \frac{1}{a} \]
Thus, we find:
\[ S = \frac{a + \frac{1}{a}}{2} \]

Now, subtracting the second equation from the first:
\[ (S + T) - (S - T) = a - \frac{1}{a} \]
This simplifies to:
\[ 2T = a - \frac{1}{a} \]
Thus, we find:
\[ T = \frac{a - \frac{1}{a}}{2} \]

### Step 5: Find \( \cos x \) and \( \sin x \)
We know:
\[ \sec x = \frac{1}{\cos x} \quad \Rightarrow \quad \cos x = \frac{1}{S} = \frac{2}{a + \frac{1}{a}} = \frac{2a}{a^2 + 1} \]

To find \( \sin x \), we use the identity:
\[ \sin^2 x + \cos^2 x = 1 \quad \Rightarrow \quad \sin^2 x = 1 - \cos^2 x \]
Calculating \( \cos^2 x \):
\[ \cos^2 x = \left(\frac{2a}{a^2 + 1}\right)^2 = \frac{4a^2}{(a^2 + 1)^2} \]
Thus:
\[ \sin^2 x = 1 - \frac{4a^2}{(a^2 + 1)^2} = \frac{(a^2 + 1)^2 - 4a^2}{(a^2 + 1)^2} \]
Expanding the numerator:
\[ (a^2 + 1)^2 - 4a^2 = a^4 + 2a^2 + 1 - 4a^2 = a^4 - 2a^2 + 1 = (a^2 - 1)^2 \]
Thus:
\[ \sin^2 x = \frac{(a^2 - 1)^2}{(a^2 + 1)^2} \]

### Step 6: Take the square root to find \( \sin x \)
Taking the square root gives:
\[ \sin x = \frac{a^2 - 1}{a^2 + 1} \]

### Final Answer
Thus, the value of \( \sin x \) is:
\[ \sin x = \frac{a^2 - 1}{a^2 + 1} \]