Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

Given,

$\tan 9^{{}^{\mathrm{o}}}-\tan {27}^{{}^{\mathrm{o}}}-\tan {63}^{{}^{\mathrm{o}}}+\tan {81}^{{}^{\mathrm{o}}}$

$=\left(\cot {81}^{°}+\tan {81}^{{}^{\mathrm{o}}}\right)-\left(\tan {27}^{°}+\cot {27}^{{}^{\mathrm{o}}}\right)$

$=\left(\tan 9^{°}+\cot 9^{°}\right)-\left(\tan {27}^{°}+\cot {27}^{°}\right)$

$=\left(\frac{\sin 9^{°}}{\cos 9^{°}}+\frac{\cos 9^{°}}{\sin 9^{°}}\right)-\left(\frac{\sin {27}^{°}}{\cos {27}^{°}}+\frac{\cos {27}^{°}}{\sin {27}^{°}}\right)$

$=\frac{2}{\sin {18}^{\mathrm{o}}}-\frac{2}{\sin {54}^{\mathrm{o}}}$

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

$=8\left(\frac{1}{\sqrt{5}-1}-\frac{1}{\sqrt{5}+1}\right)$

$=8\left(\frac{\sqrt{5}+1-\left(\sqrt{5}-1\right)}{\left(\sqrt{5}-1\right)\left(\sqrt{5}+1\right)}\right)$

$=8\left(\frac{2}{5-1}\right)=4$