Question

If cos$\alpha$+cos$\beta$+cos$\gamma$=0=sin$\alpha$+sin$\beta$+sin$\gamma$ then

cos2$\alpha$+cos2$\beta$+cos2$\gamma$ equals

• A.
• B.
• C. 0
• D. 1

Answer 1

The correct option is C

0

cos$\alpha$+cos$\beta$+cos$\gamma$=0 ......(i)

sin$\alpha$+sin$\beta$+sin$\gamma$=0 .......(ii)

Let a=cos$\alpha$+isin$\alpha$, b=cos$\beta$+isin$\beta$

c=cos$\gamma$+isin$\gamma$

$\Rightarrow$ a+b+c=0 .......(iii)

$\frac{1}{a}$+$\frac{1}{b}$+$\frac{1}{c}$

=${[cos\alpha +isin\alpha ]}^{-1}$+${[cos\beta +isin\beta ]}^{-1}$+${[cos\gamma +isin\gamma ]}^{-1}$

=cos$\alpha$-isin$\alpha$+cos$\beta$-isin$\beta$+cos$\gamma$-isin$\gamma$

$\Rightarrow$ ab+bc+ca=0 .....(iv) [by (i) and (ii)]

Squaring both sides of equations (iii),

We get $a^2$+$b^2$+$c^2$+2ab+2bc+2ca=0

or $a^2$+$b^2$+$c^2$ [by (iv)]

${[cos\alpha +isin\alpha ]}^2$+${[cos\beta +isin\beta ]}^2$+${[cos\gamma +isin\gamma ]}^2$=0

$\Rightarrow$ (cos2$\alpha$+cos2$\beta$+cos2$\gamma$) + i(sin2$\alpha$+sin2$\beta$+sin2$\gamma$)

Equating the real part to zero, we have

$\Rightarrow cos2\alpha +cos2\beta +cos2\gamma =0$