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
Aug 19, 2015

The first part:

(pq)/(1-q)

= (tanx tany cos(x + y)) / (1 - tanx tany)

Using the identities tanx = sinx/cosx and 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)