Prove that?

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Answer 1 · 2017-08-20T11:00:12

Please refer to a Proof in the Explanation.

Observe that,

#(asintheta+bcostheta)^2+(acostheta-bsintheta)^2,#

#=(a^2sin^2theta cancel(+2*a*b*sinthetacostheta)+b^2cos^2theta)+(a^2cos^2thetacancel(-2*a*b*sinthetacostheta)+b^2sin^2theta),#

#=a^2(sin^2theta+cos^2theta)+b^2(cos^2theta+sin^2theta),#

#=a^2(1)+b^2(1),#

#=a^2+b^2.#

Since, # (acostheta-bsintheta)=c.............."[Given],"# we have,

#(asintheta+bcostheta)^2+c^2=a^2+b^2,#

#:. (asintheta+bcostheta)^2=a^2+b^2-c^2,#

#rArr asintheta+bcostheta=sqrt(a^2+b^2-c^2).#

Hence, the Proof.