यदि (If) `tanalpha= (1)/(7) , sin beta = (1)/(sqrt(10))` साबित करें कि `alpha+2beta= (pi)/(4)` , जहाँ` 0 lt alpha lt(pi)/(2)` और `0 lt beta lt(pi)/(2)`

Explanation

`tan (alpha+2beta)= (tanalpha+tan2beta)/(1-tanalphatan2beta)= ((1)/(7) +tan2beta)/(1-(1)/(7)tan2beta)` ....(i)
अब, `tan2beta =(2tanbeta)/(1-tan^(2)beta)= (2.(1)/(3))/(1-(1)/(9))=(3)/(4)`
` [ tan beta gt 0` क्योंकि `0 lt beta lt pi //2 ]`
`tan 2 beta ` का मान समीकरण (i ) में रखने पर,
`tan ( alpha +2beta) = ((1)/(7) +(3)/(4))/(1-(1)/(7).(3)/(4))= (25)/(25)= 1 `
अब , `0 lt alpha lt ( pi)/(2)` और `0 lt beta lt (pi)/(2)`
`:. 0 lt 2beta lt pi `, लेकिन `tan 2beta =(3)/(4) gt 0` `:. 0 lt 2 beta lt (pi)/(2)`
अतः `0 lt alpha +2betalt pi ` `:. alpha + 2beta= ( pi)/(4)`