Maharashtra BoardEnglish MediumSTD 10MathsQuestion Bank - P24 Marks
Question
Prove that $\sin ^2 A \cdot \tan A +\cos ^2 A \cdot \cot A +2 \sin A \cdot \cos A =\tan A +\cot A$
✓
Answer
$\text { L.H.S }=\sin ^2 A \cdot \tan A+\cos ^2 A \cdot \cot A+2 \sin A \cdot \cos A$
$=\sin ^2 A \cdot \frac{\sin A }{\cos A }+\cos ^2 A \cdot \frac{\cos A }{\sin A }+2 \sin A \cdot \cos A$
$=\frac{\sin ^3 A }{\cos A }+\frac{\cos ^3 A }{\sin A }+2 \sin A \cdot \cos A$
$=\frac{\sin ^4 A +\cos ^4 A +2 \sin ^2 A \cos ^2 A }{\sin A \cos A }$
$=\frac{\left(\sin ^2 A+\cos ^2 A\right)^2}{\sin A \cos A} \ldots . .\left[\because a^2+b^2+2 a b=(a+b)^2\right]$
$=\frac{1^2}{\sin A \cos A} \quad \ldots \ldots\left[\because \sin ^2 A+\cos ^2 A=1\right]$
$=\frac{1}{\sin A \cos A }$
$=\frac{\sin ^2 A +\cos ^2 A }{\sin A \cos A } \quad \ldots \ldots\left[\because 1=\sin ^2 A +\cos ^2 A \right]$
$=\frac{\sin ^2 A}{\sin A \cos A}+\frac{\cos ^2 A}{\sin A \cos A}$
$=\frac{\sin A}{\cos A}+\frac{\cos A}{\sin A}$
$=\tan A +\cot A$
$=\text { R.H.S }$
$\therefore \sin ^2 A \cdot \tan A +\cos ^2 A \cdot \cot A +2 \sin A \cdot \cos A =\tan A +\cot A$
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