Prove-that-1-cot-23-2-1-cot-22-

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

Prove that

1 - \cot 23 ° = \frac{2}{1 - \cot 22 °}

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

c o t 45 = 1

c o t (23 + 22) = 1

\Rightarrow \frac{c o t 22 c o t 23 - 1}{c o t 23 + c o t 22} = 1

\Rightarrow c o t 23 c o t 22 = 1 + c o t 22 + c o t 23

\Rightarrow - c o t 23 (1 - c o t 22) = 1 + c o t 22

\Rightarrow - c o t 23 = \frac{1 + c o t 22}{1 - c o t 22}

\Rightarrow 1 - c o t 23 = 1 + \frac{1 + c o t 22}{1 - c o t 22}

∴ 1 - c o t 23 = \frac{2}{1 - c o t 22}

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 - (\frac{1 + \tan 23 °}{1 - \tan 23 °})

\Rightarrow 1 - \cot 22 ° = \frac{- 2 \tan 23 °}{1 - \tan 23 °}

\Rightarrow (1 - \cot 23 °) \times (1 - \cot 22 °)

= (\frac{\tan 23 ° - 1}{\tan 23 °}) \times (\frac{- 2 \tan 23 °}{1 - \tan 23 °})

= 2 (p r o v e d)