If a and b are positive quantities such that a is greater than b then the minimum value of asecx−btanx

We are given the expression:

E = a sec(x) - b tan(x)

where a > b and both a and b are positive. We need to determine the minimum value of E.

Step 1: Express in Terms of Sec and Tan Identity
We use the identity:

sec²(x) - tan²(x) = 1

Rewriting E:

E = a sec(x) - b tan(x)
= √(a² - b²) * ( a/√(a² - b²) * sec(x) - b/√(a² - b²) * tan(x) )

Define θ such that:

cos(θ) = a / √(a² - b²)
sin(θ) = b / √(a² - b²)

Since cos²(θ) + sin²(θ) = 1, we can rewrite E as:

E = √(a² - b²) * (cos(θ) sec(x) - sin(θ) tan(x))

Using the trigonometric identity:

cos(θ) sec(x) - sin(θ) tan(x) = cos(x - θ)

we get:

E = √(a² - b²) cos(x - θ)

Step 2: Finding the Minimum Value
The function cos(x - θ) varies between -1 and 1.

Thus, the minimum value occurs when cos(x - θ) = -1, giving:

E_min = √(a² - b²) * (-1)
= -√(a² - b²)

Conclusion:
The minimum value of a sec(x) - b tan(x) is -√(a² - b²).