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You are here: Experts > Science > Mathematics > Advanced Math > Prove Identies (Trig)
Expert: Ahmed Salami - 10/31/2009
Question
I've trying to prove these identities (I'm stuck). See attachment "d" & "e"
Answer
Hi Jill,
d)First, you need to remember the basic identities
sin²x + cos²x = 1
sin(x+y) = sinx.cosy + siny.cosx
and when x = y
sin(x+x) = sinx.cosx + sinx.cosx
sin2x = 2sinx.cosx
Now,
sin(x + π/4) = sinx.cos(π/4) + sin(π/4).cosx
= (1/√2)sinx + (1/√2)cosx
= (1/√2)(sinx + cosx)
sin²(x + π/4) = (1/√2)².(sinx + cosx)²
= (1/2)(sin²x + 2sinx.cosx + cos²x)
= (1/2)(sin²x + cos²x + 2sinx.cosx)
= (1/2)(1 + sin2x)
2sin²(x + π/4) = 1 + sin2x
e)What you need do here is divide all four terms on the left by cosx.cosy, i.e
[sinx.cosy + siny.cosx] / [cosx.cosy - sinx.siny]
= [(sinx.cosy/cosx.cosy) + (siny.cosx/cosx.cosy)] / [(cosx.cosy/cosx.cosy) - (sinx.siny/cosx.cosy)]
= [(sinx/cosx) + (siny/cosy)] / [1 - (sinx/cosx).(siny/cosy)]
= [tanx + tany] / [1 - tanx.tany]
Regards
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