Given `tan x = -4/3` what are the values of `(sin x + cos x - tanx )/(sec x+ csc x - cot x)`

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justaguide | College Teacher | (Level 2) Distinguished Educator

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It is given that `tan x = -4/3` . The value of `(sin x + cos x - tan x)/(sec x + csc x - cot x)` has to be determined.

Using the relation `1 + tan^2 x = sec^2x` gives two values of sec x.

`sec^2x = 1 + 16/9 = 25/9`

`sec x = +- 5/3`

  • If `sec x = -5/3`

`cos x = -3/5`

`sin x = sqrt(1 - cos^2x) = sqrt(1 - 9/25) = 4/5`

`csc x = 5/4`

`cot x = -3/4`

`(sin x + cos x - tan x)/(sec x + csc x - cot x)`

= `(4/5 - 3/5 + 4/3)/(-5/3 + 5/4 + 3/4)`

= `23/5`

  • If `sec x = 5/3`

`cos x = 3/5`

`sin x = sqrt(1 - cos^2x) = sqrt(1 - 9/25) = -4/5`

`csc x = -5/4`

`cot x = -3/4`

`(sin x + cos x - tan x)/(sec x + csc x - cot x)`

= `(-4/5 + 3/5 + 4/3)/(5/3 - 5/4 + 3/4)`

= `34/35`

The value of `(sin x + cos x - tan x)/(sec x + csc x - cot x)` is `34/35` and `23/5` .

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