Given tan x tan y = p and cos(x+y)=q .Show that sin x sin y =(pq)/(1-q) and cos(x-y)= (q(1+p))/(1-p) ?

Question

Given tan x tan y = p and cos(x+y)=q .Show that sin x sin y =(pq)/(1-q) and cos(x-y)= (q(1+p))/(1-p) ?

1 Answer

Impact of this question

2547 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2015-08-19T12:54:32

The first part:

$\frac{p q}{1 - q}$ $(pq)/(1-q)$

$= \frac{tan x tan y cos (x + y)}{1 - tan x tan y}$ $= (tanx tany cos(x + y)) / (1 - tanx tany)$

Using the identities $tan x = \frac{sin x}{cos x}$ $tanx = sinx/cosx$ and $cos (x + y) = cos x cos y - sin x sin y$ $cos(x+y) = cosxcosy - sinxsiny$ on the numerator and denominator, and cross-multiplying in the denominator:

$= ({(sinx siny)/(cancel(cosx cosy))}cancel({ cosx cosy - sinx siny})) / [(cancel(cosx cosy - sinx siny))/ (cancel(cosx cosy))]$

$= sinx siny$

The second part:

$(q (1 +p))/(1 - p)$

$= (cosx cosy - sinx siny) [(1 + tanx tany)/ (1 - tanx tany)]$

From here we can do the following:

$(1 + tanx tany)/ (1 - tanx tany) = (1+(sinxsiny)/(cosxcosy))/(1-(sinxsiny)/(cosxcosy))$

Cross-multiply to get:

$= (((cosxcosy)+(sinxsiny))/(cancel(cosxcosy)))/(((cosxcosy)-(sinxsiny))/(cancel(cosxcosy)))$

$= (cosxcosy+sinxsiny)/(cosxcosy-sinxsiny)$

Thus, we have:

$= cancel((cosx cosy - sinx siny)) [ (cosx cosy + sinx siny)/ cancel((cosx cosy - sinx siny))]$

$= cosx cosy + sinx siny$

$= cos (x - y)$

Science

Math

Humanities

... and beyond

Given tan x tan y = p and cos(x+y)=q .Show that sin x sin y =(pq)/(1-q) and cos(x-y)= (q(1+p))/(1-p) ?

1 Answer

Related questions

Impact of this question