What is the simplified value of `(1/(sec A+ tan A))^2` ?

Options

A. secA + tan A
B. sinA cosA
C. `"(1-sin A)"/"(1+sinA)"`
D. `"(1-cosA)"/"(1+cosA)"`

Answer

To simplify the expression \((1/(sec A + tan A))^2\), we will follow these steps:

### Step 1: Rewrite the expression
The expression can be rewritten as:
\[ \left(\frac{1}{\sec A + \tan A}\right)^2 \]

### Step 2: Use trigonometric identities
Recall the trigonometric identities:
\[ \sec A = \frac{1}{\cos A} \quad \text{and} \quad \tan A = \frac{\sin A}{\cos A} \]
Thus, we can rewrite \(\sec A + \tan A\):
\[ \sec A + \tan A = \frac{1}{\cos A} + \frac{\sin A}{\cos A} = \frac{1 + \sin A}{\cos A} \]

### Step 3: Substitute back into the expression
Now substitute this back into our expression:
\[ \left(\frac{1}{\frac{1 + \sin A}{\cos A}}\right)^2 \]
This simplifies to:
\[ \left(\frac{\cos A}{1 + \sin A}\right)^2 \]

### Step 4: Simplify the expression
Now we can simplify this further:
\[ \frac{\cos^2 A}{(1 + \sin A)^2} \]

### Step 5: Final expression
Thus, the simplified value of the original expression \((1/(sec A + tan A))^2\) is:
\[ \frac{\cos^2 A}{(1 + \sin A)^2} \]