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5 November, 02:51

How do I simplify (tanx/1+secx) + (1+secx/tanx)

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  1. 5 November, 03:00
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    Tan x / (1 + sec x) + (1+sec x) / tan x

    Tan x=sin x / cos x

    1 + sec x=1 + 1/cos x = (cos x+1) / cos x

    Therefore:

    tan x / (1 + sec x) = (sin x/cos x) / (cos x+1) / cos x=

    = (sin x * cos x) / [cos x * (cos x+1) ]=sin x / (Cos x+1)

    (1+sec x) / tan x=[ (cos x+1) / cos x] / (sin x/cos x) =

    =[cos x (cos x+1) ] / (sin x * cos x) = (cos x+1) / sin x

    tan x / (1 + sec x) + (1+sec x) / tan x=

    =sin x / (Cos x+1) + (cos x+1) / sin x=

    = (sin²x+cos²x+2 cos x+1) / [sin x (cos x+1) ]=

    Remember: sin²x+cos²x=1⇒ sin²x=1-cos²x

    = (1-cos²x+cos²x+2 cos x+1) / [sin x (cos x+1) ]=

    =2 cos x+2 / [sin x (cos x+1) ]=

    =2 (cos x+1) / [sin x (cos x+1) ]=

    =2 / sin x

    Answer : tan x / (1 + sec x) + (1+sec x) / tan x = 2/sin x
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