1.

If `acostheta+bsintheta=m and asintheta-bcostheta=n`, then show that ` a^(2)+b^(2)=m^(2)+n^(2)`.

Answer» Given that, `" "acostheta+bsintheta=m" "…(i)` ltBrgt and `" "asintheta-bcostheta=n" "...(ii)`
On squaring and adding of Eqs. (i) and (ii), we get
`" "m^(2)+n^(2)=(acostheta+bsintheta)^(2)+(asintheta-bcostheta)^(2)`
`" "=a^(2)cos^(2)theta+b^(2)sin^(2) theta+2ab sintheta.costheta+a^(2)sin^(2)theta+b ^(2)cos^(2)theta-2ab""sintheta*costheta`
`rArr" "m ^(2)+n^(2)=a^(2)(cos^(2)theta+sin^(2)theta) +b^(2)(sin^(2)theta+ cos^(2) theta)`
`rArr" "m^(2) +n^(2)=a^(2) +b ^(2)" "` Hence proved.


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