`sqrt((1+sintheta)/(1-sintheta))+sqrt((1-sintheta)/(1+sintheta))` is equal to

Options

A. `2costheta`
B. `2sintheta`
C. `2cottheta`
D. `2sectheta`

Answer

d

Explanation

`sqrt((1+sintheta)/(1-sintheta))+sqrt((1-sintheta)/(1+sintheta))`
`=((sqrt(1+sintheta))^(2)+(sqrt(1-sintheta))^(2))/(sqrt( 1-sin^(2)theta))`
`=(1+sintheta=1-sintheta)/(cos theta)`
`=(2)/(cos theta)=2sectheta`
Aternate shortcut method
Put `theta=30^(@)`
`sqrt((1+sin30^(@))/(1-sin30^(@)))+sqrt((1-sin30^(@))/(1+sin30^(@)))`
`rArrsqrt((1+(1)/(2))/(1-(1)/(2)))+sqrt((1-(1)/(2))/(1+(1)/(2)))`
`rArrsqrt((3)/(1))+sqrt((1)/(3))` ltbr gt `rArr(4)/sqrt(3)`
Now check with options by puting `theta=30^(@)`
`2 sec30^(@)=(2xx2)/sqrt(3)=(4)/sqrt(3)`