Consider the following statements
I. If `theta=1200^(@)`, then `(sectheta+tantheta)^(-1)` is positive.
II. If `theta = 1200^(@)`, then `("cosec"theta-cottheta)` is negative.
Which of the statements given above is / are correct ?

Options

A. I only
B. II only
C. Both I and II
D. Neither I nor II

Answer

D

Explanation

Statement - I Let `theta=1200^(@)`
Consider `(sectheta+tantheta)^(-1)=(1)/(sectheta+tantheta)`
`=(sectheta-tantheta)/((sectheta+tantheta)(sectheta-tantheta))`
`=(sectheta-tantheta)/((sec^(2)theta-tan^(2)theta))=sectheta-tantheta`
When `1200divide360`
we get remainder as `120^(@)`
Now , put `theta=120^(@)`
`rArr(sectheta+tantheta)^(-1)=sec120^(@)-tan120^(@)`
`=(1)/(cos120^(@))-tan(90^(@)+30^(@))`
`=(1)/(cos(90+30))-cot30^(@)`
`=(1)/(-sin30^(@))+sqrt(3)`
`=-2+sqrt(3)` Which is negative
Now
Statement - II :
Consider `"cosec"theta-cottheta`
`=(1)/(sin(90+30))-cot(90+30)`
`=(1)/(cos30^(@))+tan30^(@)`
`=(2)/(sqrt(3))+(1)/(sqrt(3))=(3)/(sqrt(3))`
Which os positive
Hence , both statements are incorrect.