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Question Number 177791 by cortano1 last updated on 09/Oct/22

$$\mathrm{Prove}\mathrm{that}$$$$1-\cot 23°=\frac{2}{1-\cot 22°}$$

Answered by som(math1967) last updated on 09/Oct/22

$$cot45=1$$$$cot(23+22)=1$$$$\Rightarrow \frac{cot22cot23-1}{cot23+cot22}=1$$$$\Rightarrow cot23cot22=1+cot22+cot23$$$$\Rightarrow -cot23(1-cot22)=1+cot22$$$$\Rightarrow -cot23=\frac{1+cot22}{1-cot22}$$$$\Rightarrow 1-cot23=1+\frac{1+cot22}{1-cot22}$$$$\therefore 1-cot23=\frac{2}{1-cot22}$$

Answered by blackmamba last updated on 09/Oct/22

$$\tan 22°=\frac{1-\tan 23°}{1+\tan 23°}$$$$\cot 22°=\frac{1+\tan 23°}{1-\tan 23°}$$$$\Rightarrow 1-\cot 22°=1-\left(\frac{1+\tan 23°}{1-\tan 23°}\right)$$$$\Rightarrow 1-\cot 22°=\frac{-2\tan 23°}{1-\tan 23°}$$$$\Rightarrow (1-\cot 23°)\times (1-\cot 22°)$$$$=\left(\frac{\tan 23°-1}{\tan 23°}\right)\times \left(\frac{-2\tan 23°}{1-\tan 23°}\right)$$$$=2\left(proved\right)$$

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