Question

if (cos alpha+cos beta)^2+(sin alpha+sin beta)^2 = gamma cos^2(alpha - beta/2) find gamma

Answer 1

SOLUTION

WE ARE NEEDED TO FIND GAMMA

SO LET US CONSIDER LHS FIRST,

LHS

${( \cos( \alpha ) + \cos( \beta ) )}^{2} + {( \sin( \alpha ) + \sin( \beta ) )}^{2} \\ = {(2 \cos( \frac{ \alpha + \beta }{2} ). \cos( \frac{ \alpha - \beta }{2} ) ) }^{2 } + {(2 \cos( \frac{ \alpha + \beta }{2} ) . \sin( \frac{ \alpha - \beta }{2} )) }^{2} \\ = 2 \cos( \frac{ \alpha + \beta }{2} )^{2} ( ({ \cos( \frac{ \alpha - \beta }{2} ) ) }^{2} + { \sin( \frac{ \alpha - \beta }{2} ) }^{2} ) \\ = </u></strong><strong><u>4</u></strong><strong><u> \cos( \frac{ \alpha + \beta }{2} )$

RHS

$\gamma { \cos^{2} ( \frac{ \alpha - \beta }{2} ) } = </u></strong><strong><u>4</u></strong><strong><u> { \cos}^{2} (\frac{ \alpha + \beta }{2} ) \\ = > \gamma ( \frac{ \alpha - \beta }{2} ) = </u></strong><strong><u>4</u></strong><strong><u>( \frac{ \alpha + \beta }{2} ) \\ = > \gamma = </u></strong><strong><u>4</u></strong><strong><u>(</u></strong><strong><u>\</u></strong><strong><u>frac</u></strong><strong><u>{</u></strong><strong><u>\alpha + \beta</u></strong><strong><u>}</u></strong><strong><u>{</u></strong><strong><u>\</u></strong><strong><u>alpha-</u></strong><strong><u>\</u></strong><strong><u>beta</u></strong><strong><u>}</u></strong><strong><u> </u></strong><strong><u>)</u></strong><strong><u>$