Express arcsec x in terms of arctan x with derivation.

To express arcsec(x) in terms of arctan(x), we will proceed with the following steps:

Step 1: Recall the definitions
By definition,

arcsec(x) = θ such that sec(θ) = x
arctan(y) = θ such that tan(θ) = y
Step 2: Establish the relationship
From the definition of secant:

sec(θ) = x
⇒ cos(θ) = 1/x

Since sec(θ) is defined for values where |x| ≥ 1, this identity is valid.

Step 3: Relating cosine to tangent
From the identity:

cos²(θ) + sin²(θ) = 1

Since cos(θ) = 1/x,

sin²(θ) = 1 - cos²(θ)
= 1 - (1/x)²
= 1 - 1/x²
= (x² - 1)/x²

Now, using sin(θ) = √[(x² - 1)/x²] = √(x² - 1)/x

Step 4: Express tan(θ) in terms of x
From the identity tan(θ) = sin(θ)/cos(θ),

tan(θ) = [√(x² - 1)/x] / (1/x)
= √(x² - 1)

Now let:

y = √(x² - 1)

Thus,

arctan(y) = θ
Since tan(θ) = √(x² - 1)

Step 5: Final Relationship
Since arcsec(x) = θ and arctan(√(x² - 1)) = θ,

arcsec(x) = arctan(√(x² - 1))

This is the required expression.