Giải bài 4 trang 19 sách bài tập toán 11 - Chân trời sáng tạo tập 1 — Không quảng cáo

Đề bài

Chứng minh các đẳng thức lượng giác sau:

a) $4\cos x\cos \left( {\frac{\pi }{3} - x} \right)\cos \left( {\frac{\pi }{3} + x} \right) = \cos 3x$;

b) $\frac{{\sin 2x\cos x}}{{\left( {1 + \cos x} \right)\left( {1 + \cos 2x} \right)}} = \tan \frac{x}{2}$;

c) $\sin x\left( {1 + 2\cos 2x + 2\cos 4x + 2\cos 6x} \right) = \sin 7x$;

d) $\frac{{{{\sin }^2}3x}}{{{{\sin }^2}x}} - \frac{{{{\cos }^2}3x}}{{{{\cos }^2}x}} = 8\cos 2x$.

Lời giải chi tiết

a) $4\cos x\cos \left( {\frac{\pi }{3} - x} \right)\cos \left( {\frac{\pi }{3} + x} \right)$ $= 2\cos x\left( {\cos \frac{{2\pi }}{3} + \cos 2x} \right)$

$= 2\cos x.\cos 2x + 2.\frac{{ - 1}}{2}\cos x$ $= \cos 3x + \cos x - \cos x$ $= \cos 3x$

b) $\frac{{\sin 2x\cos x}}{{\left( {1 + \cos x} \right)\left( {1 + \cos 2x} \right)}}$ $= \frac{{2\sin x{{\cos }^2}x}}{{\left( {1 + \cos x} \right)\left( {1 + 2{{\cos }^2}x - 1} \right)}}$ $= \frac{{2\sin x{{\cos }^2}x}}{{\left( {1 + \cos x} \right)2{{\cos }^2}x}}$

$= \frac{{\sin x}}{{1 + \cos x}}$ $= \frac{{2\sin \frac{x}{2}\cos \frac{x}{2}}}{{1 + 2{{\cos }^2}\frac{x}{2} - 1}}$ $= \frac{{2\sin \frac{x}{2}\cos \frac{x}{2}}}{{2{{\cos }^2}\frac{x}{2}}}$ $= \tan \frac{x}{2}$

c) $\sin x\left( {1 + 2\cos 2x + 2\cos 4x + 2\cos 6x} \right)$

$= \sin x + 2\sin x\cos 2x + 2\sin x\cos 4x + 2\sin x\cos 6x$

$= \sin x + \sin 3x - \sin x + \sin 5x - \sin 3x + \sin 7x - \sin 5x$ $= \sin 7x$

d) $\frac{{{{\sin }^2}3x}}{{{{\sin }^2}x}} - \frac{{{{\cos }^2}3x}}{{{{\cos }^2}x}}$ $= \frac{{{{\sin }^2}3x{{\cos }^2}x - {{\cos }^2}3x{{\sin }^2}x}}{{{{\sin }^2}x{{\cos }^2}x}}$ $= \frac{{{{\left( {\sin 3x\cos x} \right)}^2} - {{\left( {\cos 3x\sin x} \right)}^2}}}{{{{\sin }^2}x{{\cos }^2}x}}$

$= \frac{{\left( {\sin 3x\cos x + \cos 3x\sin x} \right)\left( {\sin 3x\cos x - \cos 3x\sin x} \right)}}{{{{\sin }^2}x{{\cos }^2}x}}$ $= \frac{{\sin 4x\sin 2x}}{{\frac{1}{4}{{\sin }^2}2x}}$

$= \frac{{4\sin 4x}}{{\sin 2x}}$ $= \frac{{8\sin 2x\cos 2x}}{{\sin 2x}}$ $= 8\cos 2x$