Question

how do i simplify (secx - cosx) / sinx?

i tried splitting the numerator up so that i had (secx / sinx) - (cosx / sinx)

and then i changed sec x to 1/ cosx so that i had

((1/cosx)/ sinx) - (cos x / sinx)

after that i get stuck

Answer 1

sec x = 1 / cos x

sec x - cos x = 1 / cos x - cos x =

1 / cos x - cos ^ 2 x / cos x = ( 1 - cos ^ 2 x ) / cos x

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Remark :

1 - cos ^ 2 x = sin ^ 2 x

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1 / cos x - cos ^ 2 x / cos x = ( 1 - cos ^ 2 x ) / cos x = sin ^ 2 x / cos x


( sec x - cos x ) / sin x = ( sin ^ 2 x / cos x ) / sin x =

sin ^ 2 x / ( sin x * cos x ) =

sin x * sin x / ( sin x * cos x ) =

sin x / cos x = tan x


( sec x / sin x ) - ( cos x / sin x ) = tan x