https://file.aiquickdraw.com/mathbot/file/1730894356201tskazap2.jpg

To solve the equation provided:

\[
\left(\frac{2}{1 + \tan^2(\alpha)}\right)^2 + \frac{15}{1 + \tan^2(\alpha)} = 4,
\]

we can proceed by simplifying the expression and solving for \( \alpha \).

### Step-by-Step Solution:

1. **Introduce a Substitution:**
   Let \( x = 1 + \tan^2(\alpha) \). This substitution uses the Pythagorean identity \( 1 + \tan^2(\alpha) = \sec^2(\alpha) \).

   With this substitution, the equation becomes:
   \[
   \left(\frac{2}{x}\right)^2 + \frac{15}{x} = 4.
   \]

2. **Simplify the Equation:**
   Expanding \( \left(\frac{2}{x}\right)^2 \) gives \( \frac{4}{x^2} \), so the equation becomes:
   \[
   \frac{4}{x^2} + \frac{15}{x} = 4.
   \]

3. **Clear the Denominator by Multiplying by \( x^2 \):**
   Multiply both sides by \( x^2 \) to eliminate the fractions:
   \[
   4 + 15x = 4x^2.
   \]

4. **Rearrange the Equation into a Standard Quadratic Form:**
   Bring all terms to one side of the equation:
   \[
   4x^2 - 15x - 4 = 0.
   \]

5. **Solve the Quadratic Equation:**
   Use the quadratic formula:
   \[
   x = \frac{-(-15) \pm \sqrt{(-15)^2 - 4 \cdot 4 \cdot (-4)}}{2 \cdot 4}.
   \]
   Simplify the expression:
   \[
   x = \frac{15 \pm \sqrt{225 + 64}}{8}.
   \]
   \[


This page provides a step-by-step solution to solve the trigonometric equation \( \left(\frac{2}{1 + \tan^2(\alpha)}\right)^2 + \frac{15}{1 + \tan^2(\alpha)} = 4 \). Key mathematical concepts include the Pythagorean identity, substitution, and solving quadratic equations. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation