If ` 4 sin theta = 3 cos theta , " then " (sec^(2) theta )/( 4 ( 1 - tan^(2) theta ))` is

Options

A. `(25)/( 16)`
B. ` (25)/( 28)`
C. ` (1)/(4)`
D. ` (16)/( 25)`

Answer

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

**Step 1:** Given the equation:
\[ 4 \sin \theta = 3 \cos \theta \]

**Step 2:** Rearranging the equation:
\[ \frac{\sin \theta}{\cos \theta} = \frac{3}{4} \]
This implies:
\[ \tan \theta = \frac{3}{4} \]

**Step 3:** Finding \( \tan^2 \theta \):
\[ \tan^2 \theta = \left(\frac{3}{4}\right)^2 = \frac{9}{16} \]

**Step 4:** Using the identity for \( \sec^2 \theta \):
\[ \sec^2 \theta = 1 + \tan^2 \theta \]
Substituting the value of \( \tan^2 \theta \):
\[ \sec^2 \theta = 1 + \frac{9}{16} \]
To add these, convert 1 to a fraction:
\[ \sec^2 \theta = \frac{16}{16} + \frac{9}{16} = \frac{25}{16} \]

**Step 5:** Now, we need to evaluate the expression:
\[ \frac{\sec^2 \theta}{4(1 - \tan^2 \theta)} \]

**Step 6:** Calculate \( 1 - \tan^2 \theta \):
\[ 1 - \tan^2 \theta = 1 - \frac{9}{16} \]
Convert 1 to a fraction:
\[ 1 - \tan^2 \theta = \frac{16}{16} - \frac{9}{16} = \frac{7}{16} \]

**Step 7:** Substitute \( \sec^2 \theta \) and \( 1 - \tan^2 \theta \) into the expression:
\[ \frac{\sec^2 \theta}{4(1 - \tan^2 \theta)} = \frac{\frac{25}{16}}{4 \cdot \frac{7}{16}} \]

**Step 8:** Simplifying the expression:
\[ = \frac{25}{16} \cdot \frac{16}{4 \cdot 7} = \frac{25}{4 \cdot 7} = \frac{25}{28} \]

Thus, the final answer is:
\[ \frac{25}{28} \]

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