Question

Prove that: cot²A - cos²A = cos²A·cot²A

Answer 1

Step-by-step explanation:

To prove:-

$cot^{2}A-cos^{2}A=cos^{2}A$ · $cot^{2}A$

Proof:-

$LHS=cot^{2}A-cos^{2}A$

⇒ $\frac{cos^{2}A}{sin^{2}A}-\frac{cos^{2}A}{1}$ $($∵ $cotA=\frac{cosA}{sinA} )$

⇒ $\frac{cos^{2}A-sin^{2}Acos^{2}A}{sin^{2}A}$

⇒ $\frac{cos^{2}A(1-sin^{2}A)}{sin^{2}A}$

⇒ $\frac{cos^{2}A(cos^{2}A)}{sin^{2}A}$ $($∵ $cos^{2}A=1-sin^{2}A)$

⇒ $\frac{cos^{4}A}{sin^{2}A}$

$RHS=cos^{2}A$ · $cot^{2}A$

⇒ $\frac{cos^{2}A}{1} (\frac{cos^{2}A}{sin^{2}A})$  $($∵ $cotA=\frac{cosA}{sinA} )$

⇒ $\frac{cos^{4}A}{sin^{2}A}$

$\frac{cos^{4}A}{sin^{2}A}=\frac{cos^{4}A}{sin^{2}A}$

$LHS=RHS$

∴ $cot^{2}A-cos^{2}A=cos^{2}A$ · $cot^{2}A$

Hence proved.