यदि ` (sin^(4) A)/a + (cos^(4) A)/b = 1/(a+b)`, तो
सिद्ध कीजिए कि -
` (sin^(8)A)/(a^(3)) + (cos^(8)A)/(b^(3))= 1/(a+b)^(3)`

Explanation

` (sin^(4)A)/a + (cos^(4)A)/b = 1/(a+b)`
` rArr ( a+b)* ((sin^(4)A)/a + (cos^(4)A)/b)= 1 = 1^(2)`
` = ( sin^(2) A + cos^(2)A)^(2)`
` rArr sin^(4) A + a/b cos^(4) A + b/a sin^(4) A + cos^(4) A `
` = sin^(4) A + cos^(4)A + 2 sin^(2) A cos^(2) A `
` rArr a/b cos^(4) A - 2 sin^(2) A cos ^(2) A + b/a sin^(4).A = 0 `
` rArr a/b cos^(4) A - 2 sin ^(2) sin^(2) A cos^(2) A + b/a sin^(4) A = 0 `
` rArr ( sqrt(a/b) cos^(2) A - sqrt(b/a)sin^(2) A)^(2)=0`
` rArr sqrt(a/b) cos^(2) A - sqrt(b/a) sin^(2) A = 0 `
` rArr sqrt(a/b) cos^(2) A = sqrt(b/a) sin^(2) A `
` rArr a/b = (sin^(2) A)/(cos^(2)A)`
` rArr (sin^(2)A)/a = (cos^(2)A)/b`
` rArr (sin^(2) A)/a = (cos^(2) A)/b = (sin^(2) A + cos^(2) A )/(a+b)`
` rArr (sin^(2)A)/a = (cos^(2)A)/b = 1/(a+b)`
` rArr sin^(2) A = a/(a+b) " और " cos^(2) A = b/(a+b)`
अब ` L.H.S = (sin^(8)A)/(a^(3)) + (cos^(8)A)/(b^(3))`
` = 1/(a^(3)) * (a/(a+b))^(4) + 1/(b^(3))* (b/(a+b))^(4)`
` = 1 /(a+b)^(3) = R.H.S`