Simplifying a Trigonometric Expression

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[mbmillermo]

@SyberMath — Thanks for all that you do. I've been learning a lot from you. I'm curious about your background. You've said that you aren't a mathematician, but you know a lot of math. Were you in engineering? Math ed? What was it? Or did you just love math and teach yourself?

[bsmith]

Third method - convert the trigonometric fractions into expressions of sec and tan.
cos(x)/(1+sin(x)) + (1+sin(x))/cos(x)
= 1/(sec(x)+tan(x)) + sec(x)+tan(x)
Now use the trigonometric conjugate sec(x)-tan(x) on the fraction
= + sec(x)+tan(x)
Then use a trigonometric version of Pythagorean theorem: sec^2(x)=tan^2(x)+1
= sec(x)-tan(x) + sec(x)+tan(x) = 2*sec(x)

[seanfraser]

cosx/(1+sinx) = cosx(1-sinx)/(1-sin^2x) = (cosx - cosxsinx)/cos^2x

(1+sinx)/cosx = (cosx + cosxsinx)/cos^2x

So the sum is

2cosx/cos^2x = 2/cosx = 2secx

[toveirenestrand]

cosx/(1 + sinx) + (1 + sinx)/cosx = cos^2x/(1 + sinx)cosx + (1 + sinx)^2/(1 + sinx)cosx = (cos^2x + 1 + 2sinx + sin^2x)/cox(1 + sinx) = 2(1 + sinx)/cos(1 + sinx) = 2/cosx.

[adamsmithson]

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