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If p sin^(3)alpha+qcos^(3)alpha=sinalpha...

If `p sin^(3)alpha+qcos^(3)alpha=sinalphacosalpha` and `p sinalpha - q cos alpha=0,` then prove that : `p^(2)+q^(2)=1`

Text Solution

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`"p sin"alpha -qcosalpha=0`
`rArr p sin"alpha=qcosalpha` ...(1)
`:. "p sin" alpha+qcos^(3)alpha=sinalphacosalpha`
`rArr("p sina"alpha)sin^(2)alpha+qcos^(3)alpha=sinalphacosalpha`
`rArr(qcosalpha)sin^(2)alpha+qcos^(3)alpha=sinalphacosalpha` [from(1)]
`rArrqcosalpha(sin^(2)alpha+cos^(2)alpha)=sinalphacosalpha`
`rArr qcosalpha=sinalphacosalpha` (`:'sin^(2)alpha+cos^(2)alpha=1`)
`rArr q=sinalpha` ...(2)
Put `q=sinalpha` in equation (1)
`"p sin"alpha=sinalphacosalpha`
`rArrp=cosalpha` ...(3)
Now , `p^(2)+q^(2)=cosalpha+sinalpha=1`
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