Simplifying the Expression: Rewrite Arcsec Function and Apply Identity and Definition

(1/. a)arcsec(u/. a)+c =

To simplify the expression, we will rewrite the arcsec function in terms of its reciprocal, the sec function

To simplify the expression, we will rewrite the arcsec function in terms of its reciprocal, the sec function.

The identity relating the arcsec function with the sec function is:

arcsec(x) = sec^(-1)(x)

Let’s rewrite the given expression using this identity:

(1/a) * sec^(-1)(u/a) + c

Next, we will simplify the expression by applying the definition of the sec^(-1) function:

sec^(-1)(x) = y if and only if sec(y) = x and 0 ≤ y ≤ π or – π/2 ≤ y ≤ π/2

Using this definition, we can rewrite the expression as:

(1/a) * sec^(-1)(u/a) + c = (1/a) * y + c

where sec(y) = u/a and 0 ≤ y ≤ π or – π/2 ≤ y ≤ π/2

Now, we need to isolate y:

Multiply both sides of sec(y) = u/a by a:
a * sec(y) = u

Take the reciprocal of both sides:
1/(a * sec(y)) = 1/u

The reciprocal of sec(y) is cos(y):
1/(a * cos(y)) = 1/u

Multiply both sides by u:
u/(a * cos(y)) = 1

Multiply both sides by (a * cos(y)):
u = a * cos(y)

Now we have a value for u in terms of y.

Substitute u with a * cos(y) in the simplified expression:
(1/a) * y + c = (1/a) * sec^(-1)(u/a) + c

(1/a) * y + c = (1/a) * sec^(-1)(a * cos(y)/a) + c

(1/a) * y + c = (1/a) * sec^(-1)(cos(y)) + c

And that’s the simplified expression for (1/a)arcsec(u/a).

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