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

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 $\sin \left(\theta\right)$, where $\sin \left(\theta\right)$ ≥ 0
and RHS = $\sin \left(\theta\right)$, where $\sin \left(\theta\right)$ ≥ 0
Thus LHS = RHS for all values ot $\theta$.

Aug 21, 2016

Square LHS and simplify - as below.

#### Explanation:

$L H S = \sqrt{\frac{\sin \theta \tan \theta}{\sec} \theta}$

Consider the square of the LHS:

$L H {S}^{2} = \frac{\sin \theta \tan \theta}{\sec} \theta$

$= \sin \theta \times \sin \frac{\theta}{\cos} \theta \times \cos \theta$

$= \sin \theta \times \sin \frac{\theta}{\cancel{\cos}} \theta \times \cancel{\cos} \theta$
(cos theta !=0 -> θ != pi/2 +npi  for all $n \in \mathbb{Z}$)

$= {\sin}^{2} \theta$

$\therefore L H {S}^{2} = {\sin}^{2} \theta$

$L H S = \pm \sin \theta = \left\mid \sin \right\mid \theta$

$L H S = R H S$
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