Question

Prove the identity:
$\frac{sinA}{1+cosA}=cosecA-cotA$ [3 MARKS]

Answer 1

Formula: 1 Mark
Proof: 2 Marks
LHS = $\frac{sinA}{1+cosA}$
Multiplying the numerator and denominator by (1-cos A), we get:
LHS = $\frac{sinA}{(1+cosA)}\times \frac{(1-cosA)}{(1-cosA)}$
$=\frac{sinA(1-cosA)}{1-co{s}^{2}A}$
$=\frac{sinA(1-cosA)}{si{n}^{2}A}$ [$\because 1-co{s}^{2}A=si{n}^{2}A$]
$=\frac{1-cosA}{sinA}=\frac{1}{sinA}-\frac{cosA}{sinA}$
$=cosecA-cotA=RHS$
Hence, proved.