How do you solve cos(2arcsin 1/4) using inverse trig. functions??!!

PLEASE HELP ME!

To solve the expression cos(2arcsin(1/4)) using inverse trigonometric functions, we first need to transform the expression using the double-angle identity for cosine.

The double-angle identity for cosine states that cos(2θ) = 1 - 2sin²(θ).

In this case, θ represents arcsin(1/4). To find the value of θ, we need to use the inverse trigonometric function arcsin.

Now, let's start by finding θ:

arcsin(1/4) = θ

To solve for θ, we will use arcsin on both sides:

sin(arcsin(1/4)) = sin(θ)

Using the property of inverse trigonometric functions where sin(arcsin(x)) = x, we have:

1/4 = sin(θ)

Now we can substitute the value of sin(θ) into the double-angle identity for cosine:

cos(2arcsin(1/4)) = 1 - 2sin²(arcsin(1/4))

Substituting sin(θ) = 1/4:

cos(2arcsin(1/4)) = 1 - 2(1/4)²

Simplifying:

cos(2arcsin(1/4)) = 1 - 2(1/16)

cos(2arcsin(1/4)) = 1 - 2/16

cos(2arcsin(1/4)) = 1 - 1/8

cos(2arcsin(1/4)) = 7/8

Therefore, cos(2arcsin(1/4)) = 7/8.