Question

cos(cos⁻¹(- 1/5)+sin⁻¹(- 1/5)) is.......,Select Proper option from the given options.
(a) 4/9
(b) 1/3
(c) 0
(d) - 1/3

Answer 1

Dear Student,

Answer: Option C (0)

Solution:

First solve using given formula's
${cos}^{ - 1} ( - x )= \pi - {cos}^{ - 1} x \\ \\ {sin}^{ - 1} ( - x) = - {sin}^{ - 1} x$
${cos}^{ - 1} ( \frac{ - 1}{5} ) = \pi -{cos}^{ - 1} ( \frac{ 1}{5} ) \\ \\ {sin}^{ - 1} ( - \frac{1}{5} ) = - {sin}^{ - 1} ( \frac{1}{5} ) \\ \\ \cos({cos}^{ - 1} ( \frac{ - 1}{5} ) +{sin}^{ - 1} ( \frac{ - 1}{5} ) ) = \\ \\ \\ \cos(\pi -{cos}^{ - 1} ( \frac{ 1}{5} ) -{sin}^{ - 1} ( \frac{1}{5} )) \\ \\ \cos(\pi -({cos}^{ - 1} ( \frac{ 1}{5} ) + {sin}^{ - 1} ( \frac{1}{5} )) \\ \\ since \: \: {sin}^{ - 1} x + {cos}^{ - 1} x = \frac{\pi}{2} \\ \\ = cos(\pi - \frac{\pi}{2} ) \\ \\ = cos \frac{\pi}{2} \\ = 0$
Hope it helps you

Answer 2

we have to find the value of cos[cos^-1(-1/5) + sin^-1(-1/5) ]

if you know a formula , you can definitely solve it just 1 or 2 second.

formula is -----> sin^-1x + cos^-1x = π/2 , where |x| < 1

in above question if we assume -1/5 = x
then, it seems like cos[cos^-1x + sin^-1x]
from above formula, cos^-1x + sin^-1x = π/2

so, cos[cos^-1x + sin^-1x] = cosπ/2 = 0

hence, option (c) is correct.