1.15: A trigonometric identity and a definite integral

4PM1/1R/November/2020 — Question 6 · 7 marks

(a) Show that
sin(A+B)+sin(A−B)=2sin A cos B.\mathrm{sin}(A+B)+\mathrm{sin}(A-B)=2\mathrm{sin}\,A\,\mathrm{cos}\,B.
(2)
(b) Hence express 2sin 7x cos x\displaystyle 2\mathrm{sin}\,7x\,\mathrm{cos}\,x in the form
sin mx+sin nx,\mathrm{sin}\,mx+\mathrm{sin}\,nx,
where m\displaystyle m and n\displaystyle n are integers, giving the value of m\displaystyle m and the value of n\displaystyle n.
(1)
(c) Use calculus to evaluate
∫0π46sin 7x cos x dx.\int_0^{\frac{\pi}{4}}6\mathrm{sin}\,7x\,\mathrm{cos}\,x\,\mathrm{d}x.
(4)