MATHEMATICS M2 QUESTION

25. In the figure, ABC is an equilateral triangle of sides x. P, Q and R are the feet if the perpendicular of A, B and C to DE respectively, DAB is a straight line and angle ADE = θ where π/ 6 < θ < π/ 2.


圖片參考：http://imgcld.yimg.com/8/n/HA00610265/o/701004020028013873372260.jpg


a) Prove that
i) PR = x cose (π/3 - θ)
ii) PQ = x cos θ
iii) QR = x cose (2π/3 - θ)

b) using the result of a) , prove that cos θ + cos θ( 2π/3 - θ) = cos (π/3 - θ)

c) Using the above results and putting θ = π/4, prove that
i) cos π/12 - sin π/12 = √ 2 /2.
ii) cos π /12 + sin π /12 = √6/2.

d) Hence, find the value of sec π /12.