tg x - tg x sin^2 x and cos x = -4/5
Since the request of the problem is vague, the only thing you can do, under the given conditions, is to evaluate the value of the expression `tan x - tan x*sin^2 x` for `cos x = -4/5` .
You should factor out `tan x ` such that:
`tan x - tan x*sin^2 x = tan x(1 - sin^2 x)`
You should remember the basic formula of trigonometry such that:
`sin^2 x + cos^2 x = 1 => cos^2 x = 1 - sin^2 x`
Hence, substituting `cos^2 x` for `1 - sin^2 x` yields:
`tan x(1 - sin^2 x) = tan x*cos^2 x`
You should remember that `tan x = sin x/cos x` such that:
`sin x/cos x*cos^2 x = sin x*cos x `
Notice that the problem provides the values of cos x, hence, using the basic formula of trigonometry yields:
`sin x = +-sqrt(1 - 16/25) => sin x = +-sqrt(9/25) => sin x = +-3/5`
Notice that the value of cosine is negative, hence, the angle x is either in the quadrant 2, where sine is positive, or in quadrant 3, where sine is negative.
`tan x - tan x*sin^2 x = +-(3/4)*(4/5) = +-12/25`
Hence, evaluating the expression, under the given conditions, yields `tan x - tan x*sin^2 x =+-12/25` .
Your question is unclear.
So let's assum that cos x=-4/5 and need to find out the value of tg x-tg x sin^2x.
sin^2 x=1-cos^2 x=1(-4/5)^2=9/25,
therefore sin x=-3/5 or 3/5.
If sin x=-3/5,
then tg x=sin x/cos x=-3/5/(-4/5)=3/4.
Hence,tg x - tg x sin^2 x=3/4-3/4*(9/16)=48/64-27/64=21/64.
If sin x=3/5,
then tg x=sin x/cos x=3/5/(-4/5)=-3/4.
Hence,tg x - tg x sin^2 x=-3/4-(-3/4)*(9/16)
Hope that helps.