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

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The trigonometric identity `tan x + cos x/(1+sin x) = 1/cos x` has to be proved.

Start with the left hand side.

`tan x + cos x/(1+sin x)`

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

= `(sin x*(1+sin x) + cos x*cos x)/(cos x*(1+sin x))`

= `(sin x + sin^2x + cos^2x)/(cos x*(1+sin x))`

Use the fact `sin^2x+cos^2x = 1`

= `(sin x + 1)/(cos x*(1+sin x))`

= `1/(cos x)`

This proves `tan x + cos x/(1+sin x) = 1/cos x`

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