How do you verify the identity #sqrt((sinthetatantheta)/sectheta)=abs(sintheta)#?

2 Answers
Aug 21, 2016

Show that LHS = RHS for all #theta#

Explanation:

LHS simplifies to #sqrt ((sin(theta).sin(theta)/cos(theta))/(1/cos(theta))#using the definitions of tan and sec.
This simplifies to #sqrt(sin(theta)^2#
and hence to #sin (theta)#, where #sin(theta)# ≥ 0
and RHS = #sin(theta)#, where #sin(theta)# ≥ 0
Thus LHS = RHS for all values ot #theta#.

Aug 21, 2016

Square LHS and simplify - as below.

Explanation:

#LHS = sqrt((sin theta tan theta)/sec theta)#

Consider the square of the LHS:

# LHS^2 = (sin theta tan theta)/sec theta#

#= sin thetaxxsin theta/cos theta xx cos theta#

#= sin thetaxxsin theta/cancel cos theta xx cancel cos theta#
(#cos theta !=0 -> θ != pi/2 +npi # for all #n in ZZ#)

#= sin^2 theta#

#:. LHS^2 = sin^2 theta#

#LHS = +- sin theta = abs sin theta#

#LHS = RHS#