If `sec theta =(5)/(4),"find the value of "((2cos theta - sin theta ))/((cot theta - tan theta ))=.`

Options

A. (12)/(7)
B. (12)/(5)
C. (1)/(7)
D. (2)/(5)

Answer

A

Explanation

Consider a `Delta ABC "in which "angle A=theta and angle B=90^(@)`.
`sec theta =("hypotenuse")/("base")=(AC)/(AB)=(5)/(4)=(5x)/(4x)("say")`
`therefore AC=5x and AB=4x, "where"x` is positive .
By pythagoras' theorem we have
`AC^(2)=AB^(2)+BC^(2)`
`implies BC^(2)=AC^(2)-AB^(2)=(5x)^(2)-(4x)^(2)=9x^(2)`
`implies BC=3x.`
`therefore cos theta =(AB)/(AC) =(4x)/(5x)=(4)/(5) , sin theta =(BC)/(AC)=(3x)/(5x)=(3)/(5),`
`cot theta =(AB)/(BC)=(4x)/(3x)=(4)/(3), and tan theta =(BC)/(AB)=(3x)/(4x)=(3)/(4).`
`therefore ((2 cos theta - sin theta))/((cot theta - tan theta))=((2xx(4)/(5)-(3)/(5)))/(((4)/(3)-(3)/(4)))=(((8)/(5)-(3)/(5)))/(((4)/(3)-(3)/(4)))=(((5)/(5)))/(((7)/(12)))=(1)/(((7)/(12)))=(12)/(7).`