Question

Prove that: cos A / (cosec A + cot A -1) + sin A / (sec A + tan A -1) = 1

Answer 1

$\frac{cosA}{cosecA+cotA-1} + \frac{sinA}{secA+tanA-1}$
$\frac{cosA.sinA}{1+cosA-sinA} + \frac{sinA.cosA}{1+sinA-cosA}$
$\frac{sinA.cosA}{1+(cosA-sinA)} + \frac{sinA.cosA}{1-(cosA-sinA)}$
$\frac{sinA.cosA-sinAcosA(cosA-sinA)+sinAcosA+sinA.cosA(cosA-sinA)}{1^2-(cosA-sinA)^2}$
[tex] \frac{2sinA.cosA}{1-(cos^2A+sin^2A-2sinA.cosA)} [/tex]
$\frac{2sinA.cosA}{1-(1-2sinAcosA)}$
$\frac{2sinA.cosA}{2sinA.cosA} = 1$

Answer 2

$\frac{cosA}{cosecA+cotA-1}+\frac{sinA}{secA+tanA-1}=1\\\\L=\frac{cosA}{\frac{1}{sinA}+\frac{cosA}{sinA}-\frac{sinA}{sinA}}+\frac{sinA}{\frac{1}{cosA}+\frac{sinA}{cosA}-\frac{cosA}{cosA}}=\frac{cosA}{\frac{1+cosA-sinA}{sinA}}+\frac{sinA}{\frac{1+sinA-cosA}{cosA}}\\\\=\frac{sinAcosA}{1+cosA-sinA}+\frac{sinAcosA}{1+sinA-cosA}\\\\=\frac{sinAcosA+sin^2AcosA-sinAcos^2A+sinAcosA+sinAcos^2A-sin^2AcosA}{1+sinA-cosA+cosA+sinAcosA-cos^2A-sinA-sin^2A+sinAcosA}$

$=\frac{2sinAcosA}{2sinAcosA+1-(cos^2A+sin^2A)}=\frac{2sinAcosA}{2sinAcosA+1-1}=\frac{2sinAcosA}{2sinAcosA}=1=R$