If `A+B=90^(@)` ,then `sqrt((tanA.tanB+tanA.cotB)/(sinA.secB)-(sin^(2)B)/(cos^(2)A))`=?

Options

A. `2tanA`
B. `tanA`
C. `tan^(2)A`
D. `2cot^(2)B`

Answer

To solve the given problem, we need to simplify the expression step by step. The expression is:

\[ \sqrt{\frac{\tan A \tan B + \tan A \cot B}{\sin A \sec B - \frac{\sin^2 B}{\cos^2 A}}} \]

Given that \( A + B = 90^\circ \), we can use the relationships between the trigonometric functions for complementary angles.

### Step 1: Substitute \( B \) with \( 90^\circ - A \)

Since \( B = 90^\circ - A \), we can replace \( B \) in the expression:

\[ \tan B = \tan(90^\circ - A) = \cot A \]
\[ \cot B = \cot(90^\circ - A) = \tan A \]
\[ \sin B = \sin(90^\circ - A) = \cos A \]
\[ \sec B = \sec(90^\circ - A) = \csc A \]

### Step 2: Rewrite the expression

Now, substituting these values into the original expression:

\[ \sqrt{\frac{\tan A \cot A + \tan A \tan A}{\sin A \csc A - \frac{\cos^2 A}{\cos^2 A}}} \]

### Step 3: Simplify the numerator

The numerator becomes:

\[ \tan A \cot A + \tan^2 A = 1 + \tan^2 A \]

Using the identity \( 1 + \tan^2 A = \sec^2 A \):

\[ = \sec^2 A \]

### Step 4: Simplify the denominator

The denominator becomes:

\[ \sin A \csc A - \frac{\cos^2 A}{\cos^2 A} = 1 - 1 = 0 \]

### Step 5: Substitute back into the expression

Now we have:

\[ \sqrt{\frac{\sec^2 A}{0}} \]

Since the denominator is zero, this expression is undefined.

### Final Answer

Thus, the expression simplifies to an undefined form due to division by zero.

---