If $\csc x = \frac{5}{4}$, then find the value of

$$\frac{5 \sin^2 x + 8 \cot^2 x}{27 an^2 x - 125 \cos^2 x}$$

Question

If $\csc x = \frac{5}{4}$, then find the value of

$$\frac{5 \sin^2 x + 8 \cot^2 x}{27 	an^2 x - 125 \cos^2 x}$$

Accepted Answer

**Solution:**

1. Given: $\csc x = \frac{5}{4}$

Recall that $\csc x = \frac{1}{\sin x}$, so:
   $$\sin x = \frac{1}{\csc x} = \frac{4}{5}$$

2. Calculate $\sin^2 x$:
   $$\sin^2 x = \left(\frac{4}{5}ight)^2 = \frac{16}{25}$$

3. Find $\cos x$ using $\sin^2 x + \cos^2 x = 1$:
   $$\cos^2 x = 1 - \sin^2 x = 1 - \frac{16}{25} = \frac{9}{25}$$
   $$\cos x = \frac{3}{5} \quad 	ext{(taking positive root as usual)}$$

4. Calculate $\cot x$ and $	an x$:
   $$\cot x = \frac{\cos x}{\sin x} = \frac{3/5}{4/5} = \frac{3}{4}$$
   $$	an x = \frac{\sin x}{\cos x} = \frac{4/5}{3/5} = \frac{4}{3}$$

5. Calculate $\cot^2 x$ and $	an^2 x$:
   $$\cot^2 x = \left(\frac{3}{4}ight)^2 = \frac{9}{16}$$
   $$	an^2 x = \left(\frac{4}{3}ight)^2 = \frac{16}{9}$$

6. Calculate $\cos^2 x$ (already found):
   $$\cos^2 x = \frac{9}{25}$$

7. Substitute all values into the expression:

Numerator:
   $$5 \sin^2 x + 8 \cot^2 x = 5 	imes \frac{16}{25} + 8 	imes \frac{9}{16} = \frac{80}{25} + \frac{72}{16}$$

Simplify each term:
   $$\frac{80}{25} = \frac{16}{5} = 3.2$$
   $$\frac{72}{16} = \frac{9}{2} = 4.5$$

Sum:
   $$3.2 + 4.5 = 7.7 = \frac{77}{10}$$

Denominator:
   $$27 	an^2 x - 125 \cos^2 x = 27 	imes \frac{16}{9} - 125 	imes \frac{9}{25}$$

Calculate each term:
   $$27 	imes \frac{16}{9} = 3 	imes 16 = 48$$
   $$125 	imes \frac{9}{25} = 5 	imes 9 = 45$$

Difference:
   $$48 - 45 = 3$$

8. Final value:
   $$\frac{\frac{77}{10}}{3} = \frac{77}{10} 	imes \frac{1}{3} = \frac{77}{30}$$

**Answer:** $\frac{77}{30}$

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