Question
Prove that $\sqrt{\frac{1+\cos\text{A}}{1-\cos\text{A}}}=(\text{cosec A}+\cot\text{A}).$
✓
Answer
$\sqrt{{\frac{1+\cos\text{A}}{1-\cos\text{A}}}}=(\text{cosec }\text{A}+\cot\text{A})$$\text{LHS}=\sqrt{\frac{1+\cos\text{A}}{1-\cos\text{A}}}$
Multiplying the numerator and denominator by $(1+\cos\text{A}),$ we have:
$\sqrt{\frac{(1+\cos\text{A})^2}{(1-\cos\text{A})(1+\cos\text{A})}}$
$=\sqrt{\frac{(1+\cos\text{A})^2}{1-\cos^2\text{A}}}$
$=\frac{1+\cos\text{A}}{\sqrt{\sin^2\text{A}}}$
$=\frac{1+\cos\text{A}}{\sin\text{A}}$
$=\frac{1}{\sin\text{A}}+\frac{\cos\text{A}}{\sin\text{A}}$
$=\text{cosec }\text{A}+\cot\text{A}$
$=\text{RHS}$
Hence proved.
Multiplying the numerator and denominator by $(1+\cos\text{A}),$ we have:
$\sqrt{\frac{(1+\cos\text{A})^2}{(1-\cos\text{A})(1+\cos\text{A})}}$
$=\sqrt{\frac{(1+\cos\text{A})^2}{1-\cos^2\text{A}}}$
$=\frac{1+\cos\text{A}}{\sqrt{\sin^2\text{A}}}$
$=\frac{1+\cos\text{A}}{\sin\text{A}}$
$=\frac{1}{\sin\text{A}}+\frac{\cos\text{A}}{\sin\text{A}}$
$=\text{cosec }\text{A}+\cot\text{A}$
$=\text{RHS}$
Hence proved.
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