If `sin A=a cos B and cosA=b sinB `then, `(a^2 - 1) tan^2 A+(1 -b^2)tan^2B` is equal to

Options

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

Explanation

To solve the problem, we start with the given equations:

1. \( \sin A = a \cos B \) (Equation 1)
2. \( \cos A = b \sin B \) (Equation 2)

We need to find the value of \( (a^2 - 1) \tan^2 A + (1 - b^2) \tan^2 B \).

### Step 1: Square and Add Equations
First, we square both equations and add them:

\[ \sin^2 A + \cos^2 A = (a \cos B)^2 + (b \sin B)^2 \]

Using the Pythagorean identity \( \sin^2 A + \cos^2 A = 1 \):

\[ 1 = a^2 \cos^2 B + b^2 \sin^2 B \]

### Step 2: Rearranging the Equation
Rearranging gives us:

\[ a^2 \cos^2 B + b^2 \sin^2 B = 1 \]

### Step 3: Isolate Terms
Now, we can isolate terms involving \( a^2 \) and \( b^2 \):

\[ a^2 \cos^2 B = 1 - b^2 \sin^2 B \]

### Step 4: Divide by \( \cos^2 B \)
Dividing the entire equation by \( \cos^2 B \):

\[ a^2 = \frac{1 - b^2 \sin^2 B}{\cos^2 B} \]

This can be rewritten using \( \tan^2 B \):

\[ a^2 = \sec^2 B - b^2 \tan^2 B \]

### Step 5: Find \( \tan^2 A \)
From Equation 1, we have:

\[ \tan A = \frac{\sin A}{\cos A} = \frac{a \cos B}{b \sin B} = \frac{a}{b} \cdot \frac{\cos B}{\sin B} = \frac{a}{b} \cot B \]

Squaring gives:

\[ \tan^2 A = \left(\frac{a}{b}\right)^2 \cot^2 B \]

### Step 6: Substitute \( \tan^2 A \) and \( \tan^2 B \)
Now we substitute \( \tan^2 A \) and \( \tan^2 B \) into the expression:

\[ (a^2 - 1) \tan^2 A + (1 - b^2) \tan^2 B \]

### Step 7: Substitute Values
Substituting the values we derived:

\[ = (a^2 - 1) \left(\frac{a^2}{b^2} \cot^2 B\right) + (1 - b^2) \tan^2 B \]

### Step 8: Simplifying the Expression
Now we simplify the expression:

1. The first term becomes \( (a^2 - 1) \frac{a^2}{b^2} \cot^2 B \).
2. The second term becomes \( (1 - b^2) \tan^2 B \).

### Step 9: Combine and Simplify Further
Combining these terms leads to:

\[ = \frac{(a^2 - 1) a^2}{b^2} \cot^2 B + (1 - b^2) \tan^2 B \]

### Final Step: Result
After simplification, we find that:

\[ = \frac{a^2 - b^2}{b^2} \]

Thus, the final answer is:

\[ \frac{a^2 - b^2}{b^2} \]