a sec theta + b tan theta = 1

a sec theta - b tan theta = 5

Then a² (b²+4)=?

To solve the system of equations given by \( a \sec \theta + b \tan \theta = 1 \) and \( a \sec \theta - b \tan \theta = 5 \), we can use a method of elimination or substitution. First, we can add and subtract these equations to isolate the variables \( a \sec \theta \) and \( b \tan \theta \).

Step 1: Add the Equations

Let's add the two equations together:

• Equation 1: \( a \sec \theta + b \tan \theta = 1 \)
• Equation 2: \( a \sec \theta - b \tan \theta = 5 \)

By adding them, we get:

\( (a \sec \theta + b \tan \theta) + (a \sec \theta - b \tan \theta) = 1 + 5 \)

Which simplifies to:

\( 2a \sec \theta = 6 \)

This leads to:

\( a \sec \theta = 3 \)

Step 2: Subtract the Equations

Now, let’s subtract the second equation from the first:

\( (a \sec \theta + b \tan \theta) - (a \sec \theta - b \tan \theta) = 1 - 5 \)

Which simplifies to:

\( 2b \tan \theta = -4 \)

Thus, we find:

\( b \tan \theta = -2 \)

Step 3: Solve for \( a \) and \( b \)

Now we have two expressions:

• From \( a \sec \theta = 3 \), we can express \( a \) as:

\( a = \frac{3}{\sec \theta} = 3 \cos \theta \)

• From \( b \tan \theta = -2 \), we can express \( b \) as:

\( b = \frac{-2}{\tan \theta} = -2 \cot \theta \)

Step 4: Substitute into the Desired Expression

Now we need to find \( a^2 (b^2 + 4) \). First, we find \( a^2 \) and \( b^2 \):

\( a^2 = (3 \cos \theta)^2 = 9 \cos^2 \theta \)

\( b^2 = (-2 \cot \theta)^2 = 4 \cot^2 \theta \)

Substituting these values into the expression:

\( b^2 + 4 = 4 \cot^2 \theta + 4 = 4 (\cot^2 \theta + 1) \)

Utilizing the identity \( \cot^2 \theta + 1 = \csc^2 \theta \), we get:

\( b^2 + 4 = 4 \csc^2 \theta \)

Final Calculation

Now substituting back into \( a^2 (b^2 + 4) \):

\( a^2 (b^2 + 4) = 9 \cos^2 \theta \cdot 4 \csc^2 \theta \)

Since \( \csc^2 \theta = \frac{1}{\sin^2 \theta} \) and \( \cos^2 \theta = 1 - \sin^2 \theta \), we can express this as:

\( = 36 \cdot \frac{\cos^2 \theta}{\sin^2 \theta} = 36 \cot^2 \theta \)

Thus, the final answer we derive from these calculations is:

\( a^2 (b^2 + 4) = 36 \cot^2 \theta \)