Question

If sin(sin inverse 1/5+cos inverse x) = 1, find x

Answer 1

Hey there !!!!

sin(sin⁻¹1/5+cos⁻¹x)=1

sin⁻¹1/5+cos⁻¹x=sin⁻¹1

But sinπ/2=1
 
       ⇒sin⁻¹1=π/2
So,

  sin⁻¹1/5+cos⁻¹x=π/2--------Equation 1

In inverse trigonometry 

    sin⁻¹t+cos⁻¹t=π/2

So, equation 1 follows the above identity

So, sin⁻¹ 1/5+cos⁻¹x=π/2

Therefore value of x will be 1/5.

Hope this helped you .....

Answer 2

sin( sin-¹ 1/5 + cos-¹ x) = 1

sin-¹ 1/5 + cos-¹ x = sin-¹(1)

cos-¹ x = sin-¹ 1 - sin-¹ 1/5

Let sin-¹ 1 = P => 1 = sinP
cosP = 0 { becoz sin x = 1 , cosx = 0
sin-¹ 1/5 = Q => 1/5 = sinQ
cosQ = 2√6/5

cos( P - Q) = cosP.cosQ + sinP. sinQ

= 0 × 2√6/5 + 1 × 1/5

= 1/5

cos( P - Q) = 1/5

( P - Q) = cos-¹( 1/5)

sin-¹ 1 - sin-¹ 1/5 = cos-¹( 1/5) , put this in equation (1)

cos-¹ x = cos-¹( 1/5)

x = 1/5 ( answer )