यदि (If) `cos theta = ( cosalpha - cos beta ) / (1-cosalphacos beta) ` साबित करें कि `tan.( theta)/(2)` का एक मान ` tan. ( alpha)/(2) cot . ( beta)/(2)` है

Explanation

`tan^(2) .( theta)/(2) = ( 1- cos theta)/( 1+ cos theta ) = ( 1- ( cos theta - cos beta)/(1-cos alphacos beta))/( 1+ ( cos alpha-cos beta)/(1- cos alphacos beta))`
`= ( 1- cos alpha cos beta - cos alpha + cos beta)/( 1- cos alpha cos beta + cos alpha - cos beta)`
` = ((1- cos alpha ) + cos beta ( 1- cos alpha))/( ( 1+ cos alpha )- cos beta( 1+ cos alpha))`
`= ((1- cos alpha ) ( 1+ cos beta ) )/( (1+ cos alpha ) ( 1- cos beta))= tan^(2). ( alpha)/( 2) cot^(2). ( beta)/(2)`
`:. tan . ( theta)/(2) = +- tan. ( alpha)/( 2) cot . ( beta)/(2)`
अतः `tan . ( theta)/(2)` का एक मान ` tan . ( alpha)/(2) cot. ( beta)/(2)` है ।