Prove that: 1+ - ( 2 +x ) 1+ + ( 2 +x ) =2 - Vidyadip

Question

Prove that: $\Big\{1+\cot\text{x}-\sec\Big(\frac{\pi}{2}+\text{x}\Big)\Big\}\Big\{1+\cot\text{x}+\sec\Big(\frac{\pi}{2}+\text{x}\Big)\Big\}=2\cot\text{x}$

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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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