Question

prove that √1+cos8/1-cos8=codec+cot8​

Answer 1

$LHS = \sqrt{ \frac{1 + cos8}{1 - cos8} } = \sqrt{ \frac{1 + 2cos^{2} ( \frac{8}{2} ) - 1}{1 - 1 + 2 {sin}^{2} (\frac{8}{2} )} } \\ = \frac{cos4}{sin4} = cot4\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ RHS = cosec8 + cot8 = \frac{1}{sin8} + \frac{cos8}{sin8} \: \: \: \: \: \: \: \: \\ = \frac{ 1 + 2cos {}^{2} ( \frac{8}{2} ) - 1}{2.sin4 .cos4} = \frac{cos4}{sin4} = cot4$

$\large{\red{\sf{★LHS=RHS}}}$

$\large{\green{\sf{★Hence\: proved}}}$