The value of ( a → × b → ) 2 + ( a → · b → ) 2 2 ( a → ) 2 ( b…

Given, $\frac{(\overrightarrow{a}\times \overrightarrow{b}{)}^2+(\overrightarrow{a}·\overrightarrow{b}{)}^2}{2\left(\overrightarrow{a}{)}^2\right(\overrightarrow{b}{)}^2}$

We know, ${\left(\overrightarrow{a}\right)}^2=\overrightarrow{a}·\overrightarrow{a}=a^2, (\overrightarrow{a}\times \overrightarrow{b}{)}^2=a^2{b}^2{\sin }^2\theta , (\overrightarrow{a}·\overrightarrow{b}{)}^2=a^2{b}^2{\cos }^2\theta$

Where, $\left|\overrightarrow{a}\right|=a, \left|\overrightarrow{b}\right|=b$

$=\frac{a^2{b}^2{\sin }^2\theta +a^2{b}^2{\cos }^2\theta }{2a^2{b}^2}$

$=\frac{a^2{b}^2({\sin }^2\theta +{\cos }^2\theta )}{2a^2{b}^2}$

$=\frac{1}{2}$