If $\overrightarrow{\mathbf{x}} + \overrightarrow{\mathbf{y}} + \overrightarrow{\mathbf{z}} = \overrightarrow{\mathbf{0}},\ \left| \overrightarrow{\mathbf{x}} ight| = \left| \overrightarrow{\mathbf{y}} ight| + \left| \overrightarrow{\mathbf{z}} ight| = 2,\$and $ heta$ is angle between $\overrightarrow{\mathbf{y}}$ and $\overrightarrow{\mathbf{z}}$ , then the value of ${cosec}^{2} heta + \cot^{2} heta\$is equal to

Question

If $\overrightarrow{\mathbf{x}} + \overrightarrow{\mathbf{y}} + \overrightarrow{\mathbf{z}} = \overrightarrow{\mathbf{0}},\ \left| \overrightarrow{\mathbf{x}} ight| = \left| \overrightarrow{\mathbf{y}} ight| + \left| \overrightarrow{\mathbf{z}} ight| = 2,\$and $	heta$ is angle between $\overrightarrow{\mathbf{y}}$ and $\overrightarrow{\mathbf{z}}$ , then the value of ${cosec}^{2}	heta + \cot^{2}	heta\$is equal to

Dr. Abubacker · Accepted Answer

(b)
Given, $\left| \overrightarrow{\mathbf{x}} ight| = \left| \overrightarrow{\mathbf{y}} ight| = \left| \overrightarrow{\mathbf{z}} ight| = 2$
$and\ \overrightarrow{\mathbf{x}} = - \overrightarrow{\mathbf{y}} - \overrightarrow{\mathbf{z}}$
$\Longrightarrow \ \ {|\overrightarrow{\mathbf{x|}}}^{2} = \left| \overrightarrow{\mathbf{y}} ight|^{2} + \left| \overrightarrow{\mathbf{z}} ight|^{2} + 2\left| \overrightarrow{\mathbf{y}} ight|\left| \overrightarrow{\mathbf{z}} ight|\cos	heta$
$\Longrightarrow 4 = 4 + 4 + 2 	imes 2 	imes 2\cos	heta$
$\Longrightarrow \cos{	heta = - \frac{1}{2}}$
$\Longrightarrow 	heta = 120{^\circ}$
Now, ${cosec}^{2}	heta + \cot^{2}	heta = {cosec}^{2}120{^\circ} + \cot^{2}120{^\circ}$
$= \left( \frac{2}{\sqrt{3}} ight)^{2} + \left( - \frac{1}{\sqrt{3}} ight)^{2} = \frac{5}{3}$

Explanation

Reviewed by Dr. Abubacker

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