Question

How do you prove `cosX / (secX - tanX) = 1 + sinX`?

Answer 1

As below.

Explanation:

To prove `cos x / (sec x - tan x) = (1 + sin x)`

L H S `= cos x / ((1/cos x) - (sin x / cos x)` as `color(blue)(sec x = 1/cos x, tan x = sin x / cos x`

`=> cos x / ((1 - sin x) / cos x)` as `color(green)(cos x` is the L C M of Denominator.

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

`=> = (1 - sin^2 x) / (1 - sin x)` as `color(blue)(cos^2x = 1 - sin^2x`

`=> ((1+ sin x) *color(red)(cancel (1 - sin x))) /color(red)(cancel (1 - sin x))`

`=> 1 + sin x`

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