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Trigonometric Functions — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSTrigonometric Functions3 Marks
Question
Prove that:
$\Big\{1+\cot\text{x}-\sec\Big(\frac{\pi}{2}+\text{x}\Big)\Big\}\Big\{1+\cot\text{x}+\sec\Big(\frac{\pi}{2}+\text{x}\Big)\Big\}=2\cot\text{x}$
✓

Answer

$\text{L.H.S}=\Big\{1+\cot\text{x}-\sec\Big(\frac{\pi}{2}+\text{x}\Big)\Big\}\Big\{1+\cot\text{x}+\sec\Big(\frac{\pi}{2}+\text{x}\Big)\Big\}$
$\{{1+\cot\text{x}-(-\text{cosec}\text{x})}\}\{1+\cot\text{x}-\text{cosec}\text{x}\}$ $\Big(\because\sec\Big(\frac{\pi}{2}+\text{x}\Big)=-\text{cosec }\text{x}\Big)$
$=\{(1+\cot)+\text{cosec}\}\{(1+\cot\text{x})-\text{cosec}\text{x}\}$
$=(1+\cot\text{x})^2-\text{cosec}^2\text{x}$
$=1+\cot^2\text{x}+2\cot\text{x}-\text{cosec}^2\text{x}$
$=\text{cosec}^2+2\cot\text{x}-\text{cosec}^2$ $(\because1+\cot^2\text{x}=\text{cosec}^2\text{x})$
$=2\cot\text{x}$
$\text{= R.H.S}$
$\text{Proved}$

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