https://file.aiquickdraw.com/mathbot/file/1725893354816dmh8kjbd.jpg

### Solution to Q.15:

Given:
\[
4x = \csc A \quad \text{and} \quad \frac{4}{x} = \cot A
\]
We need to find the value of \( 4 \left( x^2 - \frac{1}{x^2} \right) \).

From the first equation, \( 4x = \csc A \), we get:
\[
x = \frac{\csc A}{4}
\]
From the second equation, \( \frac{4}{x} = \cot A \), we have:
\[
x = \frac{4}{\cot A}
\]

Now, recall the identity \( \csc^2 A = 1 + \cot^2 A \). Let's square both expressions for \( x \):

\[
x^2 = \frac{\csc^2 A}{16}
\quad \text{and} \quad \frac{1}{x^2} = \frac{\cot^2 A}{16}
\]

Now, calculate \( x^2 - \frac{1}{x^2} \):

\[
x^2 - \frac{1}{x^2} = \frac{\csc^2 A - \cot^2 A}{16} = \frac{1}{16}
\]

Finally, multiply by 4 to find the required expression:

\[
4 \left( x^2 - \frac{1}{x^2} \right) = 4 \times \frac{1}{16} = \frac{1}{4}
\]

Thus, the value of \( 4 \left( x^2 - \frac{1}{x^2} \right) \) is \( \frac{1}{4} \).

---

### Solution to Q.16:

We are given that \( PQ \parallel RS \), and we need to prove that \( \triangle POQ \sim \triangle SOR \).

To prove that two triangles are similar, we can use the **AA similarity criterion** (two angles are equal).

1. Since \( PQ \parallel RS \), we can apply the property of corresponding angles:
   \[
   \angle PQO = \angle SRO \quad (\text{corresponding angles since PQ} \parallel \text{RS})
   \]

2. Similarly, \( \angle PQR = \angle SRO \) because these are vertically opposite angles.

Thus, by the AA similarity criterion:
\[
\triangle POQ \sim \triangle SOR
\]

### Conclusion:
Both triangles are similar by the AA criterion.

---

Let me know if you need any further clarifications, and here are some follow-up questions:

1. What is the relationship between cotangent and cosecant?
2. How does the AA similarity criterion work in general?
3. Can you prove triangle similarity using other criteria besides AA?
4. Why do parallel lines give corresponding angles?
5. What other properties of parallel lines can be used in proving similarity?

**Tip**: Always look for corresponding angles or vertically opposite angles when dealing with parallel lines intersected by a transversal.

If 4x = cosec A and 4/x = cot A, find the value of 4[x^2 - 1/x^2]. In the figure, if PQ || RS, prove that ∆POQ ~ ∆SOR.

Learn how to solve a trigonometry-based problem involving cosecant and cotangent identities, and prove the similarity of two triangles using the AA similarity criterion. This guide covers important mathematical concepts like trigonometric identities and triangle similarity, perfect for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation