Name: Systems for Measurement of Angles
Uploaded: 2019-07-19
Duration: 3 min 52 s
Description: What is a radian? Learn how in a unit circle, the angle subtended by an arc of unit length at the centre of the circle is of measure one radian. Watch the vi...

Question

If

x + y + z = x y z

, prove that

\frac{x}{1 - x^{2}} + \frac{y}{1 - y^{2}} + \frac{z}{1 - z^{2}} = \frac{4 x y z}{(1 - x^{2}) (1 - y^{2}) (1 - z^{2})}

.

Accepted Answer

To prove: $\frac{x}{1 - x^{2}} + \frac{y}{1 - y^{2}} + \frac{z}{1 - z^{2}} = \frac{x}{1 - x^{2}} + \frac{y}{1 - y^{2}} + \frac{z}{1 - z^{2}}$

Let $x = \tan α, y = \tan β, z = \tan γ$

Given: $x + y + z = x y z$

$\Rightarrow \tan α + \tan β + \tan γ = \tan α \cdot \tan β \cdot \tan γ$

$\Rightarrow \tan α + \tan β = - \tan γ + \tan α \cdot \tan β \cdot \tan y$

$\Rightarrow \tan α + \tan β = - \tan γ (1 - \tan α \cdot \tan β)$

$\Rightarrow \frac{\tan α + \tan β}{1 - \tan α \cdot \tan β} = - \tan γ$

$\Rightarrow \tan (α + β) = \tan (π - γ)$

$\Rightarrow α + β = π - γ$

$\Rightarrow 2 α + 2 β = 2 π - 2 γ$

$\Rightarrow \tan (2 α + 2 β) = \tan (2 π - 2 γ)$

$\Rightarrow \frac{\tan 2 α + \tan 2 β}{1 - \tan 2 α \tan 2 β} = - \tan 2 γ$

$\Rightarrow \tan 2 α + \tan 2 β = - \tan 2 γ (1 - \tan 2 α \tan 2 β)$

$\Rightarrow \tan 2 α + \tan 2 β = - \tan 2 γ + \tan 2 α \tan 2 β \tan 2 γ$

$\Rightarrow \tan 2 α + \tan 2 β + \tan 2 γ = \tan 2 α \tan 2 β \tan 2 γ$

$\Rightarrow \frac{2 \tan α}{1 - \tan^{2} α} + \frac{2 \tan β}{1 - \tan^{2} β} + \frac{2 \tan γ}{1 - \tan^{2} γ} = \frac{2 \tan α}{1 - \tan^{2} α} \times \frac{2 \tan β}{1 - \tan^{2} β} \times \frac{2 \tan γ}{1 - \tan^{2} γ}$

$\Rightarrow \frac{2 x}{1 - x^{2}} + \frac{2 y}{1 - y^{2}} + \frac{2 z}{1 - z^{2}} = \frac{8 x y z}{(1 - x^{2}) (1 - y^{2}) (1 - z^{2})}$

$\Rightarrow \frac{x}{1 - x^{2}} + \frac{y}{1 - y^{2}} + \frac{z}{1 - z^{2}} = \frac{4 x y z}{(1 - x^{2}) (1 - y^{2}) (1 - z^{2})}$

Hence proved

If $x + y + z = x y z$ , prove that $\frac{x}{1 - x^{2}} + \frac{y}{1 - y^{2}} + \frac{z}{1 - z^{2}} = \frac{4 x y z}{(1 - x^{2}) (1 - y^{2}) (1 - z^{2})}$ .

Important Questions on Trigonometric Functions

Learn and Prepare for any exam you want !

If x+y+z=xyz, prove that x1-x2+y1-y2+z1-z2=4xyz1-x21-y21-z2.

Important Questions on Trigonometric Functions

Learn and Prepare for any exam you want !

If $x + y + z = x y z$ , prove that $\frac{x}{1 - x^{2}} + \frac{y}{1 - y^{2}} + \frac{z}{1 - z^{2}} = \frac{4 x y z}{(1 - x^{2}) (1 - y^{2}) (1 - z^{2})}$ .