Question #1bb35

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Answer 1 · 2017-02-08T23:04:06

See below

Cross multiply:

$\frac{cos x}{1 + sin x} = \frac{1 - sin x}{cos x}$ $color(green)(cosx)/(color(red)(1+sinx)) = (color(red)(1-sinx))/color(green)(cosx)$

$\Rightarrow {cos}^{2} x = (1 + sin x) (1 - sin x)$ $implies color(green)(cos^2 x) = color(red)((1 + sin x)(1 - sin x))$

$\Rightarrow {cos}^{2} x = 1 - {sin}^{2} x$ $implies color(green)(cos^2 x) = color(red)(1 - sin^2 x)$

And that is the Pythagorean Identity :)

Answer 2 · 2017-02-09T03:50:22

$L H S = \frac{cos x}{1 + sin x}$ $LHS=cosx/(1+sinx)$

$= \frac{cos x (1 - sin x)}{(1 + sin x) (1 - sin x)}$ $=(cosx(1-sinx))/((1+sinx)(1-sinx))$

$= \frac{cos x (1 - sin x)}{1 - {sin}^{2} x}$ $=(cosx(1-sinx))/(1-sin^2x)$

$= \frac{cos x (1 - sin x)}{{cos}^{2} x}$ $=(cosx(1-sinx))/cos^2x$

$= \frac{1 - sin x}{cos x} = R H S$ $=(1-sinx)/cosx=RHS$

proved