यदि (If) `tan. (theta )/(2) = sqrt((a-b)/(a+b)) tan. ( varphi)/(2) ` साबित करें कि
`cos theta = ( a cos varphi+b)/(a+b cos varphi) `

Explanation

प्रश्न से, `tan. (theta )/(2) = sqrt((a-b)/(a+b)) tan. ( varphi)/(2)` ….(i)
अब `cos theta = ( 1- tan^(2). ( theta)/(2))/(1+ tan^(2). ( theta)/(2)) = ( 1- ( a-b)/( a+b) tan^(2). ( varphi)/(2))/(1+ ( a-b)/(a+b) tan^(2) . (varphi)/(2))`
`= ( 1-(a-b)/(a+b) . ( sin^(2).(varphi)/(2))/(cos^(2). ( varphi)/(2)))/(1+ (a-b)/(a+b). (sin^(2). (varphi)/(2))/(cos^(2). ( varphi)/(2)))=((a+b) cos^(2). ( varphi)/(2)-(a-b) sin^(2). ( varphi)/(2))/((a+b)cos^(2). ( varphi)/(2)+ (a-b) sin^(2) . ( varphi)/(2))`
`= (a( cos^(2).( varphi)/(2)- sin^(2). ( varphi)/(2))+b(cos^(2). (varphi)/(2) + sin^(2). ( varphi)/(2)))/(a(cos^(2).(varphi)/(2) +sin^(2). ( varphi)/(2))+b(cos^(2) . (varphi)/(2)-sin^(2). (varphi)/(2)))`
`= (acos varphi+b)/(a+bcos varphi)`