Question

The value of $(\tan {63}^{\circ }-\cot {63}^{\circ })$

• A. $\frac{\sqrt{5}-1}{\sqrt{5}+1}\sqrt{10+2\sqrt{5}}$
• B. $\frac{2}{\sqrt{5}+1}\left(\sqrt{10+2\sqrt{5}}\right)$
• C. $\frac{\sqrt{5}-1}{4}\sqrt{10-2\sqrt{5}}$
• D. None of these

Answer 1

Answer: (A)

$\frac{\sqrt{5}-1}{\sqrt{5}+1}\sqrt{10+2\sqrt{5}}$

Given $\tan 63-\cot 63$

$=\frac{\sin 63}{\cos 63}-\frac{\cos 63}{\sin 63}$

$=\frac{{\sin }^{2}63-{\cos }^{2}63}{\cos 63\sin 63}$

$=-\frac{({\cos }^{2}63-{\sin }^{2}63)}{\cos 63\sin 63}$

$=-\frac{2({\cos }^{2}63-{\sin }^{2}63)}{2\sin 63\cos 63}$

$=-\frac{2\cos 126}{\sin 126}$

$=-2\cot 126$

$=-2\cot (90+36)$

$=2\tan 36$

$=\frac{2\sin 36}{\cos 36}$

$=\frac{2\left(\frac{1}{2}\sqrt{\frac{5-\sqrt{5}}{2}}\right)}{\left(\frac{\sqrt{5}+1}{4}\right)}$

$=\frac{\sqrt{5}-1}{\sqrt{5}+1}\sqrt{10+2\sqrt{5}}$

Ans is $Opt:\left[A\right]$