Following previous blog post, we have shown that if
If A,B,C are three angles from a triangle, i.e. A+B+C=π, then we should have known the following equality holds.
tanA+tanB+tanC=tanAtanBtanC
On this blog post, we now try to generate another relation between A,B and C for cotangent functions:
1.
A+B+C=π
A+B=π−C
Taking the sine on both sides of the equation:
sin(A+B)=sin(π−C)
sinAcosB+cosAsinB=sinπcosC−cosπsinC
sinAcosB+cosAsinB=(0)(cosC)−(−1)(sinC)
sinAcosB+cosAsinB=sinC---(1)
Taking the cosine on both sides of the equation:
cos(A+B)=cos(π−C)
cosAcosB−sinAsinB=cosπcosC+sinπsinC
cosAcosB−sinAsinB=(−1)(cosC)+(0)(sinC)
cosAcosB−sinAsinB=−cosC---(2)
Dividing (2) by (1), we see that we have:
cosAcosB−sinAsinBsinAcosB+cosAsinB=−cosCsinC
Dividing the numerator and denominator on the left by sinAsinB, we get:
cosAcosBsinAsinB−sinAsinBsinAsinBsinAcosBsinAsinB+cosAsinBsinAsinB=−cotC
cotAcotB−1cotB+cotA=−cotC
cotAcotB−1=(−cotC)(cotB+cotA)
cotAcotB−1=−cotCcotB−cotCcotA)
\displaystyle \color{yellow}\bbox[5px,green]{\cot A \cot B+\cot B\cot C+\cot A\cot C=1}
2.
A+B+C=\pi
\displaystyle \frac{A}{2}+\frac{B}{2}=\frac{\pi}{2}-\frac{C}{2}
Taking the sine on both sides of the equation:
\displaystyle \sin \left(\frac{A}{2}+\frac{B}{2}\right)=\sin \left(\frac{\pi}{2}-\frac{C}{2}\right)
\displaystyle \sin \frac{A}{2} \cos \frac{B}{2}+\cos \frac{A}{2}\sin \frac{B}{2}=\sin\frac{\pi}{2} \cos \frac{C}{2}-\cos \frac{\pi}{2} \sin \frac{C}{2}
\displaystyle \sin \frac{A}{2} \cos \frac{B}{2}+\cos \frac{A}{2}\sin \frac{B}{2}=(1)\cos \frac{C}{2}-(0)\sin \frac{C}{2}
\displaystyle \sin \frac{A}{2} \cos \frac{B}{2}+\cos \frac{A}{2}\sin \frac{B}{2}=\cos \frac{C}{2}---(3)
Taking the cosine on both sides of the equation:
\displaystyle \cos \left(\frac{A}{2}+\frac{B}{2}\right)=\cos \left(\frac{\pi}{2}-\frac{C}{2}\right)
\displaystyle \cos \frac{A}{2} \cos \frac{B}{2}-\sin \frac{A}{2}\sin \frac{B}{2}=\cos \frac{\pi}{2} \cos \frac{C}{2}+\sin \frac{\pi}{2} \sin \frac{C}{2}
\displaystyle \cos \frac{A}{2} \cos \frac{B}{2}-\sin \frac{A}{2}\sin \frac{B}{2}=(0) \cos \frac{C}{2}+(1) \sin \frac{C}{2}
\displaystyle \cos \frac{A}{2} \cos \frac{B}{2}-\sin \frac{A}{2}\sin \frac{B}{2}= \sin \frac{C}{2}---(4)
Dividing (3) by (4), we see that we have:
\displaystyle \frac{\sin \frac{A}{2} \cos \frac{B}{2}+\cos \frac{A}{2}\sin \frac{B}{2}}{\cos \frac{A}{2} \cos \frac{B}{2}-\sin \frac{A}{2}\sin \frac{B}{2}}=\frac{\cos \frac{C}{2}}{\sin \frac{C}{2}}
\displaystyle \frac{\sin \frac{A}{2} \cos \frac{B}{2}+\cos \frac{A}{2}\sin \frac{B}{2}}{\cos \frac{A}{2} \cos \frac{B}{2}-\sin \frac{A}{2}\sin \frac{B}{2}}=\cot \frac{C}{2}
Dividing the numerator and denominator on the left by \sin \frac{A}{2}\sin \frac{B}{2}, we get:
\displaystyle \frac{\frac{\sin \frac{A}{2} \cos \frac{B}{2}}{\sin \frac{A}{2}\sin \frac{B}{2}}+\frac{cos \frac{A}{2}\sin \frac{B}{2}}{\sin \frac{A}{2}\sin \frac{B}{2}}}{\frac{\cos \frac{A}{2} \cos \frac{B}{2}}{\sin \frac{A}{2}\sin \frac{B}{2}}-\frac{\sin \frac{A}{2}\sin \frac{B}{2}}{\sin \frac{A}{2}\sin \frac{B}{2}}}=\cot \frac{C}{2}
\displaystyle \frac{\cot \frac{B}{2}+\cot \frac{A}{2}}{\cot \frac{A}{2} \cot \frac{B}{2}-1}=\cot \frac{C}{2}
\displaystyle \cot \frac{B}{2}+\cot \frac{A}{2}=\cot \frac{A}{2} \cot \frac{B}{2}\cot \frac{C}{2} -\cot \frac{C}{2}
\displaystyle \color{yellow}\bbox[5px,purple]{\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}=\cot \frac{A}{2} \cot \frac{B}{2}\cot \frac{C}{2} }
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