Prove that sin x-ysin x-y-tan x-tan ytan x-tan y

To prove nbsp;sin (x+y)sin (x y)=tan x+tan ytan x tan y R.HS=tan x+tan ytan x tan y=sin xcos x+sin ycos ysin xcos x sin ycos y nbsp; nbsp; nbsp; n ...

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Standard XII

Mathematics

Solving Simultaneous Trigonometric Equations

prove that si...

Question

prove that $\frac{sin (x + y)}{sin (x - y)} = \frac{tan x + tan y}{tan x - tan y}$

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Solution

to prove $\frac{sin (x + y)}{sin (x - y)} = \frac{tan x + tan y}{tan x - tan y}$

$R . H S = \frac{tan x + tan y}{tan x - tan y} = \frac{\frac{sin x}{cos x} + \frac{sin y}{cos y}}{\frac{sin x}{cos x} - \frac{sin y}{cos y}}$

$= \frac{\frac{sin x cos y + sin y cos x}{cos x cos y}}{\frac{sin x cos y - sin y cos x}{cos x cos y}}$

$= \frac{sin x cos y + sin y cos x}{sin x cos y - sin y cos x} = \frac{sin (x + y)}{sin (x - y)}$ =L.H.S

Hence proved

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Solving Simultaneous Trigonometric Equations

Standard XII Mathematics

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prove that sin(x+y)sin(x−y)=tanx+tanytanx−tany

to prove sin(x+y)sin(x−y)=tanx+tanytanx−tany R.HS=tanx+tanytanx−tany=sinxcosx+sinycosysinxcosx−sinycosy =sinxcosy+sinycosxcosxcosysinxcosy−sinycosxcosxcosy =sinxcosy+sinycosxsinxcosy−sinycosx=sin(x+y)sin(x−y) =L.H.S Hence proved

prove that $\frac{sin (x + y)}{sin (x - y)} = \frac{tan x + tan y}{tan x - tan y}$