সৃজনশীলঃ ‘…’ কুমিল্লা বোর্ড ২০২৩ – সংযুক্ত কোনের ত্রিকোনমিতিক অনুপাত, উচ্চতর গণিত, উচ্চতর গণিত ১ম পত্র (এইচএসসি)

প্রদত্ত রাশি,

$=\frac{\tan {42}^{\circ }\tan {78}^{\circ }}{\cot 6^{\circ }\cot {66}^{\circ }}$

$=\tan 6^{\circ }\tan {42}^{\circ }\tan {66}^{\circ }\tan {78}^{\circ }$

$=\frac{\sin {66}^{\circ }\tan \sin {42}^{\circ }\sin {66}^{\circ }\sin {78}^{\circ }}{\cos 6^{\circ }\cos {42}^{\circ }\cos {66}^{\circ }\cos {78}^{\circ }}$

$=\frac{2\sin {66}^{\circ }\sin 6^{\circ }2\sin {78}^{\circ }\sin {42}^{\circ }}{2\cos {66}^{\circ }\cos 6^{\circ }\cdot 2\cos {78}^{\circ }\cos {42}^{\circ }}$

$=\frac{(\cos {60}^{\circ }-\cos {72}^{\circ })(\cos {36}^{\circ }-\cos {120}^{\circ })}{(\cos {72}^{\circ }+\cos {66}^{\circ })(\cos {120}^{\circ }+\cos {36}^{\circ })}$

$=\frac{(\frac{1}{2}-\cos {72}^{\circ })(\cos {36}^{\circ }+\frac{1}{2})}{(\cos {72}^{\circ }+\frac{1}{2})(\cos {36}^{\circ }-\frac{1}{2})}$

$=\frac{(1-2\cos {72}^{\circ })(2\cos {36}^{\circ }+1)}{(2\cos {72}^{\circ }+1)(2\cos {36}^{\circ }-1)}$

$=\frac{1-2\cos ({90}^{\circ }-{78}^{\circ })(2\cos {36}^{\circ }+1)}{1+2\cos ({96}^{\circ }-{18}^{\circ })(2\cos {36}^{\circ }-1)}$

$=\frac{(1-2\sin {18}^{\circ })(2\cos {36}^{\circ }+1)}{(1+2\sin {18}^{\circ })(2\cos {36}^{\circ }-1)}$

$=\frac{1-2\times \frac{\sqrt{5}-1}{4}}{1+2\times \frac{\sqrt{5}-1}{4}\times \frac{2\times \frac{\sqrt{5}+1}{4}+1}{2\times \frac{\sqrt{5}+1}{4}-1}}$

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

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

$=\frac{3-\sqrt{5}}{\sqrt{5}+1}\times \frac{3+\sqrt{5}}{\sqrt{5}-1}$

$=\frac{9-5}{5-1}$

$=\frac{4}{4}$

$=\frac{1}{1}$

অন্যভাবে,

প্রদত্ত রাশি, $=\frac{\tan {42}^{\circ }\tan {78}^{\circ }}{\cot 6^{\circ }\cot {66}^{\circ }}$

$=\tan 6^{\circ }\tan {42}^{\circ }\tan {66}^{\circ }\tan {78}^{\circ }$

$=\tan 6^{\circ }\tan ({60}^{\circ }-{18}^{\circ })\tan ({60}^{\circ }+6^{\circ })\tan ({60}^{\circ }+{18}^{\circ })\frac{\tan {18}^{\circ }}{\tan {18}^{\circ }}$

$=\frac{\tan 6^{\circ }\tan ({60}^{\circ }+6^{\circ })}{\tan {18}^{\circ }}\times \{\tan {18}^{\circ }\tan ({60}^{\circ }+{18}^{\circ })\tan ({60}^{\circ }-{18}^{\circ })\}$

$=\frac{\tan 6^{\circ }\tan (66+6^{\circ })}{\tan {18}^{\circ }}\times \tan (3\times {18}^{\circ })$

$\frac{=\tan 6^{\circ }\tan ({60}^{\circ }+6^{\circ })\tan {54}^{\circ }}{\tan {18}^{\circ }}$

$=\frac{\tan 6^{\circ }\tan (60+6^{\circ })\tan ({60}^{\circ }-6^{\circ })}{\tan {18}^{\circ }}$

$=\frac{\tan (3\times 6^{\circ })}{\tan {18}^{\circ }}=\frac{\tan {18}^{\circ }}{\tan {18}^{\circ }}=1$

$[\because \tan x\cdot \tan ({60}^{\circ }+x)$.

$\tan ({60}^{\circ }-x)$

$=\tan 3x]$