Question

Question in attachment plz solve it . its urgent

Answer 1

Step-by-step explanation:

We have,

$\cos^{ - 1} \bigg( \frac{ \cos(x) + \cos(y) }{1 + \cos(x) \cos(y) } \bigg)$

$= \tan^{ - 1} \bigg \{\frac { \sqrt{1 -( \frac{ \cos(x) + \cos(y) }{1 + \cos(x) \cos(y) }} )^{2} }{\frac{ \cos(x) + \cos(y) }{1 + \cos(x) \cos(y) }} \bigg \}$

$= \tan^{ - 1} \bigg \{\frac { \sqrt{(1 + \cos(x) \cos(y) ) ^{2} -( \cos(x) + \cos(y) } )^{2} }{ \cos(x) + \cos(y) } \bigg \}$

$= \tan^{ - 1} \bigg \{\frac { \sqrt{1 + \cos ^{2} (x) \cos ^{2} (y) + 2 \cos(x) \cos(y) - \cos ^{2} (x) - \cos^{2} (y) - 2 \cos(x) \cos(y) } }{ \cos(x) + \cos(y) } \bigg \}$

$= \tan^{ - 1} \bigg \{\frac { \sqrt{1 + \cos ^{2} (x) \cos ^{2} (y) - \cos ^{2} (x) - \cos^{2} (y) } }{ \cos(x) + \cos(y) } \bigg \}$

$= \tan^{ - 1} \bigg \{\frac { \sqrt{1 - \cos ^{2} (x)+ \cos ^{2} (x) \cos ^{2} (y) - \cos^{2} (y) } }{ \cos(x) + \cos(y) } \bigg \}$

$= \tan^{ - 1} \bigg \{\frac { \sqrt{(1 - \cos ^{2} (x) )- \cos ^{2} (y)(1 - \cos ^{2} (x) ) } }{ \cos(x) + \cos(y) } \bigg \}$

$= \tan^{ - 1} \bigg \{\frac { \sqrt{(1 - \cos ^{2} (x) )(1- \cos ^{2} (y)) } }{ \cos(x) + \cos(y) } \bigg \}$

$= \tan^{ - 1} \bigg \{\frac { \sqrt{\sin^{2} (x) \sin^{2} (y) } }{ \cos(x) + \cos(y) } \bigg \}$

$= \tan^{ - 1} \bigg \{\frac { \sin (x) \sin(y) }{ \cos(x) + \cos(y) } \bigg \}$

$= \tan^{ - 1} \bigg \{\frac { 4\sin ( \frac{x}{2} ) \sin( \frac{y}{2} ) \cos( \frac{x}{2} ) \cos( \frac{y}{2} ) }{ \cos(x) + \cos(y) } \bigg \}$

$= \tan^{ - 1} \bigg \{\frac { 2\tan( \frac{x}{2} ) \tan( \frac{y}{2} ) }{ \frac{\cos(x) + \cos(y) }{2 \cos ^{2} ( \frac{x}{2} ) \cos^{2} ( \frac{y}{2} ) } } \bigg \}$

$= \tan^{ - 1} \bigg \{\frac { 2\tan( \frac{x}{2} ) \tan( \frac{y}{2} ) }{ \frac{2\cos^{2} ( \frac{x}{2} ) - 1 + 2 \cos ^{2} ( \frac{y}{2} ) - 1 }{2 \cos ^{2} ( \frac{x}{2} ) \cos^{2} ( \frac{y}{2} ) } } \bigg \}$

$= \tan^{ - 1} \bigg \{\frac { 2\tan( \frac{x}{2} ) \tan( \frac{y}{2} ) }{ \sec^{2} ( \frac{y}{2} ) + \sec^{2} ( \frac{x}{2} ) - \sec^{2} ( \frac{x}{2} ) \sec ^{2} ( \frac{y}{2} ) } \bigg \}$

$= \tan^{ - 1} \bigg \{\frac { 2\tan( \frac{x}{2} ) \tan( \frac{y}{2} ) }{ 1 + \tan^{2} ( \frac{y}{2} ) + 1 + \tan^{2} ( \frac{x}{2} ) - (1 + \tan^{2} ( \frac{x}{2} ) )(1 + \tan ^{2} ( \frac{y}{2} ) ) } \bigg \}$

$= \tan^{ - 1} \bigg \{\frac { 2\tan( \frac{x}{2} ) \tan( \frac{y}{2} ) }{ 1 + \tan^{2} ( \frac{y}{2} ) + 1 + \tan^{2} ( \frac{x}{2} ) - 1 - \tan^{2} ( \frac{x}{2} ) - \tan ^{2} ( \frac{y}{2} ) - \tan^{2} ( \frac{x}{2} ) \tan ^{2} ( \frac{y}{2} ) } \bigg \}$

$= \tan^{ - 1} \bigg \{\frac { 2\tan( \frac{x}{2} ) \tan( \frac{y}{2} ) }{ 1 - \tan^{2} ( \frac{x}{2} ) \tan ^{2} ( \frac{y}{2} ) } \bigg \}$

$= 2\tan^{ - 1} \bigg\{\tan( \frac{x}{2} ) \tan( \frac{y}{2} ) \bigg \}$