InterviewSolution
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InterviewSolution
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Evaluate: i) `tan{1/2cos^(-1)sqrt(5)/3}`, ii) `tan{2tan^(-1)1/5-pi/4}` |
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Answer» i) Let `cos^(-1)sqrt(5)/3=theta`. Then, `costheta=sqrt(5)/3`. `therefore tan{1/2cos^(-1)sqrt(5)/3}=tantheta/2` `=sqrt((1-costheta)/(1+costheta))=sqrt((1-sqrt(5)/3)/(1+sqrt(5)/3))=sqrt((3-sqrt(5))/(3+sqrt(5))` `=sqrt((3-sqrt(5))/(3+sqrt(5)) xx (3-sqrt(5))/(3+sqrt(5)) = (3-sqrt(5))/sqrt(9-5)` `=(3-sqrt(5))/sqrt(4)=(3-sqrt(5))/(2)` ii) `tan{2tan^(-1)1/5-pi/4}` `=tan{tan^(-1)(2 xx 1/5)/(1-1/25))-tan^(-1)1}` `=tan{tan^(-1)(2/5 xx 25/24)-tan^(-1)1}`. `=tan{tan^(-1)5/12-tan^(-1)1}` `=tan{tan^(-1)(5/12-1)/(1+5/12)}=tan{tan^(-1)(-7/17)}=-7/17`. |
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