If `sintheta+costheta=aandsectheta+cosectheta=b`, then the value of `b(a^(2)-1)` is equal to

Options

A. 2a
B. 3a
C. 0
D. 2ab

Answer

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

1. \( \sin \theta + \cos \theta = a \)
2. \( \sec \theta + \csc \theta = b \)

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

### Step 1: Express \( a^2 \)

We know that:
\[ a = \sin \theta + \cos \theta \]
Now, squaring both sides:
\[ a^2 = (\sin \theta + \cos \theta)^2 = \sin^2 \theta + \cos^2 \theta + 2\sin \theta \cos \theta \]
Using the Pythagorean identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ a^2 = 1 + 2\sin \theta \cos \theta \]

### Step 2: Express \( b \)

Next, we express \( b \):
\[ b = \sec \theta + \csc \theta = \frac{1}{\cos \theta} + \frac{1}{\sin \theta} \]
Finding a common denominator:
\[ b = \frac{\sin \theta + \cos \theta}{\sin \theta \cos \theta} = \frac{a}{\sin \theta \cos \theta} \]

### Step 3: Substitute \( a^2 - 1 \)

Now, we can find \( a^2 - 1 \):
\[ a^2 - 1 = 2\sin \theta \cos \theta \]

### Step 4: Calculate \( b(a^2 - 1) \)

Now we substitute \( b \) and \( a^2 - 1 \) into \( b(a^2 - 1) \):
\[ b(a^2 - 1) = \left(\frac{a}{\sin \theta \cos \theta}\right)(2\sin \theta \cos \theta) \]
The \( \sin \theta \cos \theta \) terms cancel out:
\[ b(a^2 - 1) = 2a \]

### Final Result

Thus, the value of \( b(a^2 - 1) \) is:
\[ \boxed{2a} \]

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