Question

Prove that:

(i) $tan{5}^{\circ }tan{25}^{\circ }tan{30}^{\circ }tan{65}^{\circ }tan{85}^{\circ }=\frac{1}{\sqrt{3}}$

(ii) $cot{12}^{\circ }cot{38}^{\circ }cot{52}^{\circ }cot{60}^{\circ }cot{78}^{\circ }=\frac{1}{\sqrt{3}}$

(iii) $cos{15}^{\circ }cos{35}^{\circ }cosec{55}^{\circ }cos{60}^{\circ }cosec{75}^{\circ }=\frac{1}{2}$

(iv) $cos{1}^{\circ }cos{2}^{\circ }cos{3}^{\circ }...cos{180}^{\circ }=0$

(v) ${\left(\frac{sin{49}^{\circ }}{cos{41}^{\circ }}\right)}^2+{\left(\frac{cos{41}^{\circ }}{sin{49}^{\circ }}\right)}^2=2$

Answer 1

(i) $tan{5}^{\circ }tan{25}^{\circ }tan{30}^{\circ }tan{65}^{\circ }tan{85}^{\circ }=\frac{1}{\sqrt{3}}LHS=(tan{5}^{\circ }tan{25}^{\circ }tan{30}^{\circ }tan{65}^{\circ }tan{85}^{\circ }=tan({90}^o–{85}^o)\times tan({90}^o–65v)\times \frac{1}{\sqrt{3}}\times \frac{1}{cot{65}^o}\times \frac{1}{cot{85}^o}=cot{85}^o\times cot{65}^o\times \frac{1}{\sqrt{3}}\times \frac{1}{cot{65}^o}\times \frac{1}{cot{85}^o}=\frac{1}{\sqrt{3}}=RHS$

(ii) $cot{12}^{\circ }cot{38}^{\circ }cot{52}^{\circ }cot{60}^{\circ }cot{78}^{\circ }=\frac{1}{\sqrt{3}}LHS=cot{12}^{\circ }cot{38}^{\circ }cot{52}^{\circ }cot{60}^{\circ }cot{78}^{\circ }=tan({90}^o–{12}^o)\times tan({90}^o–{38}^o)\times cot{52}^o\times \frac{1}{\sqrt{3}}\times cot{78}^o=\frac{1}{\sqrt{3}}\times tan{78}^o\times tan{52}^o\times cot{52}^o\times cot{78}^o=\frac{1}{\sqrt{3}}\times tan{78}^o\times tan{52}^o\times \frac{1}{tan{52}^o}\times \frac{1}{tan{78}^o}=\frac{1}{\sqrt{3}}=RHS$

(iii) $cos{15}^{\circ }cos{35}^{\circ }cosec{55}^{\circ }cos{60}^{\circ }cosec{75}^{\circ }=\frac{1}{2}LHS=cos{15}^{\circ }cos{35}^{\circ }cosec{55}^{\circ }cos{60}^{\circ }cosec{75}^{\circ }=cos({90}^{\circ }–{75}^{\circ })\times cos({90}^{\circ }–{55}^{\circ })\times \frac{1}{sin{55}^{\circ }}\times \frac{1}{2}\times \frac{1}{sin{75}^{\circ }}=sin{75}^{\circ }\times sin{55}^{\circ }\times \frac{1}{sin{55}^{\circ }}\times \frac{1}{2}\times \frac{1}{sin{75}^{\circ }}=\frac{1}{2}=RHS$

(iv) $cos{1}^{\circ }cos{2}^{\circ }cos{3}^{\circ }...cos{180}^{\circ }=0LHS=cos{1}^{\circ }cos{2}^{\circ }cos{3}^{\circ }...cos{180}^{\circ }=cos{1}^{\circ }cos{2}^{\circ }cos{3}^{\circ }\dots \times cos{90}^{\circ }\times \dots cos{180}^{\circ }=cos{1}^{\circ }cos{2}^{\circ }cos{3}^{\circ }\dots \times 0\times \dots cos{180}^{\circ }=0=RHS$

(v) $(\frac{sin{49}^{\circ }}{cos{41}^{\circ }}{)}^2+(\frac{cos{41}^{\circ }}{sin{49}^{\circ }}{)}^2=2LHS=(\frac{sin{49}^{\circ }}{cos{41}^{\circ }}{)}^2+(\frac{cos{41}^{\circ }}{sin{49}^{\circ }}{)}^2=(\frac{cos({90}^{\circ }–{49}^{\circ })}{cos{41}^{\circ }}{)}^2+(\frac{cos{41}^{\circ }}{cos({90}^{\circ }–{49}^{\circ })}{)}^2=(\frac{cos{41}^{\circ }}{cos{41}^{\circ }}{)}^2+(\frac{cos{41}^{\circ }}{cos{41}^{\circ }}{)}^2=1^2+1^2=1+1=2=RHS$