By: Tao Steven Zheng (鄭濤)

【Problem】

In the "Treatise on the Chord and Sine", the Persian astronomer and mathematician Jamshid al-Kashi (1380 - 1429) used the trigonometric identity to calculate? accurate to 18 decimal places (). Use the trigonometric identities and ? to prove that .

【Solution】

Use the angle sum formula and make the substitution .

Substitute the double-angle formulas for sine and cosine into the above formula:?

Subsequently,

Since ,

【Additional Problem】

Using the same trigonometric identities, prove that