यदि ` p sin ^(3) alpha + q cos ^(3) alpha = sin alpha cos alpha " और " p sin alpha - q cos alpha = 0 ` तो सिद्ध कीजिए कि ` p^(2) + q^(2) = 1 `.

Explanation

` p sin alpha - q cos alpha = 0 `
`rArr p sin alpha = q cos alpha ` …(1)
` therefore p sin ^(3) alpha + q cos ^(3) alpha = sin alpha cos aloha `
` rArr (p sin alpha) sin^(2) alpha + q cos^(3) alpha = sin alpha cos alpha `
` rArr (q cos alpha ) sin^(2) alpha + q cos ^(3) alpha = sin alpha cos alpha " "`[(1) से ]
`rArr q cos alpha (sin^(2) alpha + cos^(2) alpha) = sin alpha cos alpha `
` rArr q cos alpha = sin alpha cos alha" "(because sin^(2) alpha + cos ^(2) alpha = 1 `)
`rArr q = sin alpha `...(2)
` q = sin alpha ` समीकरण (1) में रखने पर
` p sin alpha = sin alpha cos alpha `
`rArr p = cos alpha ` ...(3)
अब
, ` p^(2) + q^(2) = cos ^(2) alpha + sin^(2) alpha = 1 `