Question

Cos(sin^-1(-4/5))+sin(tan^-1(3/4))+cos(cosec^-1(5/3))​

Answer 1

We have to find the value of cos(sin¯¹(-4/5)) + sin(tan¯¹(3/4)) + cos(cosec¯¹(5/3))

solution : we know, sin¯¹(-x) = -sin¯¹x

so, sin¯¹(-4/5) = -sin¯¹(4/5)

now let sin¯¹(4/5) = A

⇒sinA = 4/5

so cosA = √(5² - 4²)/5 = 3/5

⇒cos¯¹(3/5) = A

now, sin¯¹(-4/5) = -cos¯¹(3/5)

now cos(-cos¯¹(3/5)) = 3/5 [because cos¯¹(-x) = cos¯¹x]

similarly, sin(tan¯¹(3/4))

tan¯¹(3/4) = B

⇒tanB = 3/4

⇒sinB = 3/√(3² + 4²) = 3/5

⇒B = tan¯¹(3/4) = sin¯¹(3/5)

so, sin(sin¯¹(3/5)) = 3/5

and cos(cosec¯¹(5/3))

cosec¯¹(5/3) = C

⇒cosecC = 5/3

⇒cosC = √(5² - 3²)/5 = 4/5

⇒C = cosec¯¹(5/3) = cos¯¹(4/5)

so cos(cos¯¹(4/5)) = 4/5

Therefore, cos(sin¯¹(-4/5)) + sin(tan¯¹(3/4)) + cos(cosec¯¹(5/3)) = 3/5 + 3/5 + 4/5 = 10/5 = 2