यदि (If)` sin alpha + sin beta = a ` तथा ` cos alpha + cosbeta) = b ` साबित करें `tan. ( alpha-beta)/(2) = +- sqrt((4-a^(2) -b^(2))/(a^(2) +b^(2)))`

यदि (If)` sin alpha + sin beta = a ` तथा ` cos alpha + cos

1.	यदि (If)` sin alpha + sin beta = a ` तथा ` cos alpha + cosbeta) = b ` साबित करें `tan. ( alpha-beta)/(2) = +- sqrt((4-a^(2) -b^(2))/(a^(2) +b^(2)))`
Answer» प्रश्न से, `sin alpha + sin beta = a` …(1) तथा ` cos alpha + cos beta = b ` …(2) या, ` cos^(2) alpha + cos^(2) beta + 2 cos alpha cos beta + sin^(2) alpha +sin^(2) beta + 2 sinalpha sin beta = b^(2) + a^(2)` या, ` ( cos^(2) a + sin^(2) alpha) + ( cos^(2) beta+ sin^(2) beta)+ 2 ( cos alpha cos beta +sin alpha sin beta ) =a^(2) +b^(2)` या, ` 2+2 cos ( alpha - beta) = a^(2) + b^(2)` या, `cos ( alpha - beta) = ( a^(2) + b^(2) -2)/(2)` अब, `tan. ( alpha - beta)/( 2) +- sqrt((1-cos ( alpha-beta))/(1+cos ( alpha -beta)))` `= +- sqrt((1-(a^(2) + b^(2) - 2)/(2))/(1+(a^(2) + b^(2) -2)/(2)))= +-sqrt((4-a^(2)-b^(2))/(a^(2)+b^(2)))`