If `cos theta =(3)/(5) ,"find the value of "((5"cosec "theta- 4 tan theta )/(sectheta + cot theta)).`

Options

A. `24/29`
B. `1/29`
C. `8/29`
D. `11/29`

Answer

D

Explanation

Let us draw a `Delta ABC "in which "angle B=90^(@).`
Let `angle A=theta^(@).`
then ,`cos theta =(AB)/(AC)=(3)/(5).`
Let `AB=3K and Ac=5k` , where k is positive .
By pythagoras' theorem , we have
`AC^(2)=AB^(2)+BC^(2)`
`implies BC^(2)=AC^(2)-AB^(2)=(5k)^(2)-(3k)^(2)=25k^(2)-9k^(2)=16k^(2)`
`implies BC = sqrt(16k^(2))=4k.`
`therefore sectheta =(1)/(cos theta )=(5)/(3), tan theta =(BC)/(AB)=(4k)/(3k)=(4)/(3),`
`cot theta =(1)/(tan theta )=(3)/(4) , and "cosec " theta =(AC)/(BC)=(5k)/(4k) =(5)/(4).`
`therefore ((5"cosec " theta - 4 tan theta )/(sec theta +cot theta))=((5xx(5)/(4)-4xx(4)/(3)))/(((5)/(3)+(3)/(4)))=(((25)/(4)-(16)/(3)))/(((5)/(3)+(3)/(4)))`
`=((75-64)/(12))/((20+9)/(12))=((11)/(12)xx(12)/(29))=(11)/(29).`