If $$\frac{\sec heta - an heta}{\sec heta + an heta} = \frac{1}{9}$$ and $$ heta$$ lies in the first quadrant, find the value of $$\frac{\sin heta - an^2 heta}{\sin heta + an^2 heta}$$.

Question

If $$\frac{\sec 	heta - 	an 	heta}{\sec 	heta + 	an 	heta} = \frac{1}{9}$$ and $$	heta$$ lies in the first quadrant, find the value of $$\frac{\sin 	heta - 	an^2 	heta}{\sin 	heta + 	an^2 	heta}$$.

Accepted Answer

**Solution:**

1. Given:
   $$\frac{\sec 	heta - 	an 	heta}{\sec 	heta + 	an 	heta} = \frac{1}{9}$$
   and $$	heta$$ is in the first quadrant.

2. Cross-multiply to get:
   $$9(\sec 	heta - 	an 	heta) = \sec 	heta + 	an 	heta$$

3. Expanding:
   $$9\sec 	heta - 9	an 	heta = \sec 	heta + 	an 	heta$$

4. Rearranging terms:
   $$9\sec 	heta - \sec 	heta = 9	an 	heta + 	an 	heta$$
   $$8\sec 	heta = 10	an 	heta$$

5. Simplify:
   $$4\sec 	heta = 5	an 	heta$$

6. Using identities:
   $$\sec 	heta = \frac{1}{\cos 	heta}, \quad 	an 	heta = \frac{\sin 	heta}{\cos 	heta}$$

Substitute:
   $$4 	imes \frac{1}{\cos 	heta} = 5 	imes \frac{\sin 	heta}{\cos 	heta}$$

7. Multiply both sides by $$\cos 	heta$$:
   $$4 = 5 \sin 	heta$$

8. Solve for $$\sin 	heta$$:
   $$\sin 	heta = \frac{4}{5}$$

9. Since $$	heta$$ is in the first quadrant, $$\cos 	heta$$ is positive. Use Pythagorean identity:
   $$\cos^2 	heta = 1 - \sin^2 	heta = 1 - \left(\frac{4}{5}ight)^2 = 1 - \frac{16}{25} = \frac{9}{25}$$

So,
   $$\cos 	heta = \frac{3}{5}$$

10. Calculate $$	an 	heta$$:
    $$	an 	heta = \frac{\sin 	heta}{\cos 	heta} = \frac{\frac{4}{5}}{\frac{3}{5}} = \frac{4}{3}$$

11. Now evaluate the required expression:
    $$\frac{\sin 	heta - 	an^2 	heta}{\sin 	heta + 	an^2 	heta} = \frac{\frac{4}{5} - \left(\frac{4}{3}ight)^2}{\frac{4}{5} + \left(\frac{4}{3}ight)^2} = \frac{\frac{4}{5} - \frac{16}{9}}{\frac{4}{5} + \frac{16}{9}}$$

12. Find common denominators:
    - Numerator:
      $$\frac{4}{5} - \frac{16}{9} = \frac{36}{45} - \frac{80}{45} = -\frac{44}{45}$$
    - Denominator:
      $$\frac{4}{5} + \frac{16}{9} = \frac{36}{45} + \frac{80}{45} = \frac{116}{45}$$

13. Divide numerator by denominator:
    $$\frac{-\frac{44}{45}}{\frac{116}{45}} = -\frac{44}{45} 	imes \frac{45}{116} = -\frac{44}{116} = -\frac{11}{29}$$

**Answer:** $$-\frac{11}{29}$$ (Option B)

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