Question

How do you use the double-angle identities to find cot(2x) if cos x= -15/17 and csc x is less than 0?

Answer 1

`161/240`

Explanation:

Given
`cosx =-15/17 and cscx<0->sinx<0`
`"both "cosx and sinx " being " <0 , color(blue)(" x is in 3rd quadrant")`

So `tanx>0`

`sinx =-sqrt(1-cos^2x)=-sqrt(1-(-15/17)^2)=-8/17`

`tanx = sinx/cosx=(-8/17)/(-15/17)=8/15`

Now

`cot2x=1/tan(2x)=(1-tan^2x)/(2tanx)=(1-(8/15)^2)/(2*8/15)`
`=161/15^2xx15/16=161/240`