Question

If $A+B+C={180}^{\circ }$, then the value of $(cotB+cotC)(cotC+cotA)(cotA+cotB)$ will be

• A. sec A sec B sec C
• B. cosec A cosec B cosec C
• C. tan A tan B tan C
• D. 1

Answer 1

The correct option is B cosec A cosec B cosec C

$(cotB+cotC)(cotC+cotA)(cotA+cotB)$

$=(\frac{cosB}{sinB}+\frac{cosC}{sinC})(\frac{cosC}{sinC}+\frac{cosA}{sinA})(\frac{cosA}{sinA}+\frac{cosB}{sinB})$

$=\frac{sin(B+C)}{sinBsinC}.\frac{sin(C+A)}{sinCsinA}.\frac{sin(A+B)}{sinAsinB}$

$=\frac{sin({180}^{\circ }-A)}{sinBsinC}.\frac{sin({180}^{\circ }-B)}{sinCsinA}.\frac{sin({180}^{\circ }-C)}{sinAsinB}$

$=\frac{sinAsinBsinC}{sinBsinC.sinCsinA.sinAsinB}$

$=\frac{1}{sinAsinBsinC}$