Question

Prove $\cot \frac{\pi }{24}=\sqrt{2}+\sqrt{3}+\sqrt{4}+\sqrt{6}$ ?

Answer 1

Converting Cot(π/24) into degrees by multiplying the angle by 180/π we can get

Cot(π/24)= cot 7.5

cotA= (1+cos2A)/sin2A

We can write

Cot (15/2)= (1+cos15°)/sin15°……………(1)

Now

cos15°= cos (45°-30°)

=cos45.cos30+sin45.sin30

=1/√2.√3/2+1/√2.1/2

=> cos 15°= (1/4) (√6 + √2)

Similarly sin15° = (1/4) (√6 - √2)

Putting these values in eq. (1)

Cot (15/2)= [1 + (1/4) (√6 + √2)] / [(1/4) (√6 - √2)]

= (4 + √6 + √2) / (√6 - √2)

On rationalisation

Cot (15/2)= [(4 + √6 + √2) * (√6 + √2)] / [(√6 - √2) * (√6 + √2)]

= (4√6 + 6 + 2√3 + 4√2 + 2√3 + 2) / 4

= (4√2 + 4√3 + 8 + 4√6) / 4

= √2 + √3 + √4 + √6.