Question

If sin⁻¹ 1/x = 2tan⁻¹1/7+cos⁻¹3/5, then x=.......,Select Proper option from the given options.
(a) 24/117
(b) 7/3
(c) 125/117
(d) - 117/44

Answer 1

Dear Student,

Answer: option c ( 125/117)

Solution:

1) First convert 2tan⁻¹1/7 into sin⁻¹x,
Then cos⁻¹3/5 to sin⁻¹x,

2) Apply sin⁻¹x+sin⁻¹y formula

3) sin⁻¹x cancels from both side,hence solve for x

$2 {tan}^{ - 1} x = {sin}^{ - 1} ( \frac{2x}{1 + {x}^{2} } ) \\ \\ 2 {tan}^{ - 1} ( \frac{1}{7}) = {sin}^{ - 1} ( \frac{ \frac{2}{7} }{1 + { (\frac{1}{7}) }^{2} } ) \\ \\ = {sin}^{ - 1} ( \frac{7}{25} ) \\ \\ {cos}^{ - 1} x = {sin}^{ - 1} ( \sqrt{1 - {x}^{2} } ) \\ \\ {cos}^{ - 1} \frac{3}{5} = {sin}^{ - 1} ( \sqrt{1 - \frac{9}{25} } ) \\ = {sin}^{ - 1} ( \frac{4}{5} ) \\ \\ {sin}^{ - 1} x + {sin}^{ - 1} y = \\ {sin}^{ - 1} (x \sqrt{1 - {y}^{2} } + y \sqrt{1 - {x}^{2} } ) \\ \\ {sin}^{ - 1} \frac{7}{25} + {sin}^{ - 1} \frac{4}{5} = \\ {sin}^{ - 1} ( \frac{7}{25} \sqrt{1 - \frac{16}{25} } + \frac{4}{5} \sqrt{1 - \frac{49}{652} } ) \\ \\ = {sin}^{ - 1} ( \frac{117}{125} ) \\ {sin}^{ - 1} (\frac{1}{x} ) = {sin}^{ - 1} ( \frac{117}{125} ) \\ \\ \frac{1}{x} = \frac{117}{125} \\ \\ x = \frac{125}{117}$

hope it helps you