Here is the extracted mathematical problem from the image:

Given: $P = \csc \beta - 1, \quad Q = \csc \beta + 1, \quad R = \sec \alpha - \tan \alpha$

(ক)

Given $x = \tan \theta$, $y = \cot \theta$, $z = \sin \theta$, and $x = \frac{5}{12}$, find $z$.

(খ)

Prove that: $\sqrt{\frac{Q}{P}} = \frac{1}{\sec \beta - \tan \beta}$

(গ)

If $R = \frac{1}{x}$, determine the value of $\csc \alpha$.

Would you like me to solve these step by step? Here are five related questions to deepen understanding:

1. How do you derive $\sin \theta$ from $\tan \theta$?
2. What is the relation between secant and tangent in terms of sine and cosine?
3. How do you simplify expressions involving cosecant and secant?
4. What is the geometric interpretation of the given trigonometric functions?
5. How does rationalization help in simplifying square root expressions?

Tip:

Remember that fundamental trigonometric identities like $1 + \tan^2 \theta = \sec^2 \theta$ and $1 + \cot^2 \theta = \csc^2 \theta$ are key to solving such problems efficiently.

Let me know which part you'd like me to solve first! 😊