Question #5a859

1 Answer
sente
Aug 14, 2016

We will start from the left hand side and show that it equals the right hand side. To do so, we will use the following identities:

  • #tan(x) = sin(x)/cos(x)#
  • #sec(x) = 1/cos(x)#
  • #sec^2(x) - 1 = tan^2(x)#

Note that the third identity may be derived from #sin^2(x) + cos^2(x) = 1# by dividing each side by #cos^2(x)# and then subtracting #1# from each side.

Proceeding:

#LHS = tan^2(u) - sin^2(u)#

#= (sin(u)/cos(u))^2-sin^2(u)#

#= sin^2(u)/cos^2(u)-sin^2(u)#

#=sin^2(u)(1/cos^2(u) - 1)#

#=sin^2(u)(sec^2(u)-1)#

#=sin^2(u)tan^2(u) = RHS# ∎

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