Question

if alpha and beta are the solutions of acos theta+bcos theta =c prove that cos(alpha + beta)=a^2-b^2/a^2+b^2

Answer 1

atan@+bsec@=c

bsec@=c-atan@

squaring bothsides

b^2sec^2@=(c-atan@)^2

b^2(1+tan^2@)=c^2+a^2tan^2@-2actan@

(b^2-a^2)tan^2@+2actan@+b^2-c^2=0

since alpha and beta are the roots of the eq so

tan(alpha)+tan(beta)=-2ac/(b^2-a^2)

tan(alpha).tan(beta)=(b^2-c^2)/(b^2-a^2)

tan(alpha+beta)={tan(alpha)+tan(beta)}/{1-tan(alpha).tan(beta)}

=[-2ac/(b^2-a^2)]/[1-(b^2-c^2)/(b^2-a^2)]

=-2ac/(c^2-a^2)

=2ac/(a^2-c^2) Ans.

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