Question

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

Answer 1

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`.

Answer 2

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`