1.63: Multiple-Angle Identities and Integration

4PM1/2R/June/2019 — Question 10 · 15 marks

(a) Use the formula for cos⁡(A+B)\displaystyle \cos(A+B) to show that
cos⁡2A=2cos⁡2A−1.\cos 2A=2\cos^2 A-1.
(2)
(b) Show that
cos⁡4A=8cos⁡4A−8cos⁡2A+1.\cos 4A=8\cos^4 A-8\cos^2 A+1.
(4)
(c) Solve the equation
cos⁡2(θ4+π24)[cos⁡2(θ4+π24)−1]=−116,0⩽θ<2π.\cos^2\left(\frac{\theta}{4}+\frac{\pi}{24}\right) \left[\cos^2\left(\frac{\theta}{4}+\frac{\pi}{24}\right)-1\right] =-\frac{1}{16}, \qquad 0\leqslant\theta<2\pi.
Give your answers in terms of π\displaystyle \pi.
(5)
Given that
f(A)=4cos⁡4A−4cos⁡2A+1,f(A)=4\cos^4 A-4\cos^2 A+1,
(d) using calculus, find the exact value of
∫π/6π/2f(A) dA.\int_{\pi/6}^{\pi/2} f(A)\,\mathrm{d}A.
Give your answer in the form aπ−bc\displaystyle a\pi-b\sqrt{c}, where a\displaystyle a and b\displaystyle b are fractions in their lowest terms and c\displaystyle c is a prime number.
(4)