Find the value of `(sqrt(cosec^(2)A-1))/(cot A+tanA)`?

Options

A. `sin^(2)A`
B. `cosAsinA`
C. `cos^(2)A`
D. `secAcosecA`

Answer

To solve the expression \(\frac{\sqrt{\csc^2 A - 1}}{\cot A + \tan A}\), we will follow these steps:

### Step 1: Simplify the numerator
We start with the numerator \(\sqrt{\csc^2 A - 1}\). We know from trigonometric identities that:
\[ \csc^2 A = 1 + \cot^2 A \]
Thus, we can rewrite the numerator:
\[ \sqrt{\csc^2 A - 1} = \sqrt{\cot^2 A} \]
Since \(\sqrt{\cot^2 A} = |\cot A|\), we can assume \(\cot A\) is positive in the context of this problem:
\[ \sqrt{\csc^2 A - 1} = \cot A \]

### Step 2: Simplify the denominator
Next, we simplify the denominator \(\cot A + \tan A\):
\[ \tan A = \frac{1}{\cot A} \]
Thus, we can rewrite the denominator:
\[ \cot A + \tan A = \cot A + \frac{1}{\cot A} \]
To combine these, we find a common denominator:
\[ \cot A + \frac{1}{\cot A} = \frac{\cot^2 A + 1}{\cot A} \]

### Step 3: Substitute into the expression
Now we substitute the simplified numerator and denominator back into the expression:
\[ \frac{\sqrt{\csc^2 A - 1}}{\cot A + \tan A} = \frac{\cot A}{\frac{\cot^2 A + 1}{\cot A}} = \frac{\cot^2 A}{\cot^2 A + 1} \]

### Step 4: Use the identity
We know from the Pythagorean identity that:
\[ \cot^2 A + 1 = \csc^2 A \]
Thus, we can rewrite our expression as:
\[ \frac{\cot^2 A}{\csc^2 A} \]

### Step 5: Simplify further
Since \(\csc^2 A = \frac{1}{\sin^2 A}\) and \(\cot^2 A = \frac{\cos^2 A}{\sin^2 A}\), we have:
\[ \frac{\cot^2 A}{\csc^2 A} = \frac{\frac{\cos^2 A}{\sin^2 A}}{\frac{1}{\sin^2 A}} = \cos^2 A \]

### Final Answer
Thus, the value of the expression \(\frac{\sqrt{\csc^2 A - 1}}{\cot A + \tan A}\) is:
\[ \cos^2 A \]