Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7). Assume that
0 < 𝜃 < 𝜋/2.

1
x2
,   x = 2 tan(𝜃)

To simplify the expression \( \frac{1}{x^2} \) given \( x = 2 \tan(\theta) \) using trigonometric substitution, let's follow these steps:

1. Substitute \( x = 2 \tan(\theta) \) into the expression \( \frac{1}{x^2} \).
   
   \[
   \frac{1}{x^2} = \frac{1}{(2 \tan(\theta))^2}
   \]

2. Simplify the expression inside the denominator:
   
   \[
   \frac{1}{x^2} = \frac{1}{4 \tan^2(\theta)}
   \]

3. To further simplify, use the trigonometric identity \( \sec^2(\theta) = 1 + \tan^2(\theta) \), which implies:
   
   \[
   \tan^2(\theta) = \sec^2(\theta) - 1
   \]

4. Substitute \( \tan^2(\theta) \) in terms of \( \sec(\theta) \):
   
   \[
   \frac{1}{x^2} = \frac{1}{4 (\sec^2(\theta) - 1)}
   \]

Now, we have the expression in terms of trigonometric functions, simplified to:

\[
\frac{1}{x^2} = \frac{1}{4 (\sec^2(\theta) - 1)}
\]

Would you like further details on the trigonometric identities used, or have questions?

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Here are 5 related questions that expand on this concept:

1. How would we simplify if we had \( x = 3 \sin(\theta) \) instead?
2. Can this approach apply to integrals involving trigonometric substitution?
3. What other trigonometric identities might simplify similar expressions?
4. How would we handle limits as \( \theta \to \pi/2 \) in such substitutions?
5. How is this substitution method applied in calculus for solving differential equations?

**Tip:** Remember that trigonometric substitution is often used to simplify expressions involving roots and powers, especially when they appear in integration.

Make the indicated trigonometric substitution in the given algebraic expression and simplify (see Example 7). Assume that 0 < θ < π/2. 1/x^2, x = 2 tan(θ)

Simplify the expression 1/x^2 using trigonometric substitution with x = 2 tan(θ). Follow a detailed step-by-step process, including the use of key trigonometric identities like sec^2(θ) = 1 + tan^2(θ). Suitable for students in Grades 11-12.

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