Solving Trigonometry Relation Psin60° = Qcosec45°

This problem asks us to find the value of a specific algebraic expression involving variables P and Q, given a relationship between them that uses trigonometric functions. We need to use the known values of trigonometric functions for standard angles.

Step 1: Understand the Given Trigonometric Equation

The given equation is:

Psin60° = Qcosec45°

Here, P and Q are variables, and we are given a relationship involving the sine of 60 degrees and the cosecant of 45 degrees.

Step 2: Recall Standard Trigonometric Values

We need the values for sin(60°) and csc(45°):

• sin(60°) = $\frac{\sqrt{3}}{2}$
• csc(45°) = $\frac{1}{\sin(45°)}$. Since sin(45°) = $\frac{1}{\sqrt{2}}$ (or $\frac{\sqrt{2}}{2}$), then csc(45°) = $\sqrt{2}$.

Step 3: Substitute Values and Find the Ratio P/Q

Substitute the trigonometric values back into the given equation:

P $\left(\frac{\sqrt{3}}{2}\right)$ = Q $(\sqrt{2})$

Our goal is to find the value of $\frac{P^2 + Q^2}{P^2 - Q^2}$. To do this, it's helpful to find the ratio $\frac{P}{Q}$ or $\frac{P^2}{Q^2}$.

Rearranging the equation to find $\frac{P}{Q}$:

$\frac{P}{Q} = \frac{\sqrt{2}}{\frac{\sqrt{3}}{2}}$

$\frac{P}{Q} = \sqrt{2} \times \frac{2}{\sqrt{3}} = \frac{2\sqrt{2}}{\sqrt{3}}$

Now, let's find $\frac{P^2}{Q^2}$ by squaring both sides:

$\frac{P^2}{Q^2} = \left(\frac{2\sqrt{2}}{\sqrt{3}}\right)^2 = \frac{(2\sqrt{2})^2}{(\sqrt{3})^2} = \frac{4 \times 2}{3} = \frac{8}{3}$

Step 4: Evaluate the Expression $\frac{P^2 + Q^2}{P^2 - Q^2}$

We need to calculate $\frac{P^2 + Q^2}{P^2 - Q^2}$.

To use the ratio $\frac{P^2}{Q^2}$ we found, we can divide both the numerator and the denominator of the expression by $Q^2$ (assuming $Q \neq 0$):

$\frac{\frac{P^2}{Q^2} + \frac{Q^2}{Q^2}}{\frac{P^2}{Q^2} - \frac{Q^2}{Q^2}} = \frac{\frac{P^2}{Q^2} + 1}{\frac{P^2}{Q^2} - 1}$

Now, substitute the value $\frac{P^2}{Q^2} = \frac{8}{3}$:

$\frac{\frac{8}{3} + 1}{\frac{8}{3} - 1}$

Simplify the numerator and the denominator:

Numerator: $\frac{8}{3} + 1 = \frac{8}{3} + \frac{3}{3} = \frac{8+3}{3} = \frac{11}{3}$

Denominator: $\frac{8}{3} - 1 = \frac{8}{3} - \frac{3}{3} = \frac{8-3}{3} = \frac{5}{3}$

Now, perform the division:

$\frac{\frac{11}{3}}{\frac{5}{3}} = \frac{11}{3} \times \frac{3}{5} = \frac{11}{5}$

Conclusion for Psin60° = Qcosec45°

Therefore, the value of the expression $\frac{P^2 + Q^2}{P^2 - Q^2}$ is $\frac{11}{5}$.