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
Truong-Son N. · Chandra S.
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)#

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