Chứng minh rằng:
----- Nội dung dịch tự động từ ảnh -----
Bài 5. CMR:
a) \( \cos a \sin (b - c) + \cos b \sin (c - a) + \cos c \sin (a - b) = 0 \)
b) \( 2 \sin \left(\frac{\pi}{4} + a\right) \sin \left(\frac{\pi}{4} - a\right) = \cos 2a \)
c) \( \sin a + \cos a = \sqrt{2} \cos \left(\frac{\pi}{4} - a\right) \)
d) \( \cos 5a \cos 4a = \cos a [1 - \cos 2a + \cos 4a - \cos 6a + \cos 8a] \)
Bài 6: Rút gọn biểu thức
a) \( C = 4 \cos \frac{x}{3} - \cos \frac{\pi - x}{3} \)
b) \( D = \cos^2 4 + \cos^2 (a + b) - 2 \cos a \cos b \cos (a + b) \)
c) \( \frac{\sin 2a + \sin 5a - \sin 3a}{1 + \cos 2 - 2 \sin^2 2a} \)
d) \( D = \frac{\sin x + \sin 4x + \sin 7x}{\cos x \cos 4x + \cos 7x} \)
Bài 7: Rút gọn
\[ A = \frac{1 - 2 \cos 2a}{\sqrt{3} + 2 \sin 2a}; \, B = \frac{\sqrt{2} - 2 \cos 2a}{\sqrt{2} + 2 \cos 2a}; \, C = \frac{1 - 2 \cos a}{1 + 2 \cos a} \]
\[ D = \frac{\sqrt{3} - 2 \cos 3a}{\sqrt{3} + 2 \cos 3a} \]
Bài 8: Rút gọn
\[ A = \frac{2 ( \sin 2a + 2 \cos^2 a - 1 )}{\cos a - \sin a - \cos 3a + \sin 3a}; \]
\[ B = \frac{\sqrt{2} - \sin a - \cos a}{\sin a - \cos a} \]
Bài 9. CMR:
a) \( \cos 4a + 4 \cos 2a + 3 = 8 \cos^4 a \)
b) \( 3 - 4 \cos 2a + \cos 4a = 8 \sin^4 a \)
c) \( \frac{1 + \cos a + \cos 2a + \cos 3a}{2 \cos^2 a + \cos a - 1} = 2 \cos a \)
d) \( 1 + 4 \cos a + 6 \cos 2a + 4 \cos 3a + \cos 4a = 16 \cos^2 2a \cos^4 \frac{a}{2} \)
Bài 5. CMR:
a) \( \cos a \sin (b - c) + \cos b \sin (c - a) + \cos c \sin (a - b) = 0 \)
b) \( 2 \sin \left(\frac{\pi}{4} + a\right) \sin \left(\frac{\pi}{4} - a\right) = \cos 2a \)
c) \( \sin a + \cos a = \sqrt{2} \cos \left(\frac{\pi}{4} - a\right) \)
d) \( \cos 5a \cos 4a = \cos a [1 - \cos 2a + \cos 4a - \cos 6a + \cos 8a] \)
Bài 6: Rút gọn biểu thức
a) \( C = 4 \cos \frac{x}{3} - \cos \frac{\pi - x}{3} \)
b) \( D = \cos^2 4 + \cos^2 (a + b) - 2 \cos a \cos b \cos (a + b) \)
c) \( \frac{\sin 2a + \sin 5a - \sin 3a}{1 + \cos 2 - 2 \sin^2 2a} \)
d) \( D = \frac{\sin x + \sin 4x + \sin 7x}{\cos x \cos 4x + \cos 7x} \)
Bài 7: Rút gọn
\[ A = \frac{1 - 2 \cos 2a}{\sqrt{3} + 2 \sin 2a}; \, B = \frac{\sqrt{2} - 2 \cos 2a}{\sqrt{2} + 2 \cos 2a}; \, C = \frac{1 - 2 \cos a}{1 + 2 \cos a} \]
\[ D = \frac{\sqrt{3} - 2 \cos 3a}{\sqrt{3} + 2 \cos 3a} \]
Bài 8: Rút gọn
\[ A = \frac{2 ( \sin 2a + 2 \cos^2 a - 1 )}{\cos a - \sin a - \cos 3a + \sin 3a}; \]
\[ B = \frac{\sqrt{2} - \sin a - \cos a}{\sin a - \cos a} \]
Bài 9. CMR:
a) \( \cos 4a + 4 \cos 2a + 3 = 8 \cos^4 a \)
b) \( 3 - 4 \cos 2a + \cos 4a = 8 \sin^4 a \)
c) \( \frac{1 + \cos a + \cos 2a + \cos 3a}{2 \cos^2 a + \cos a - 1} = 2 \cos a \)
d) \( 1 + 4 \cos a + 6 \cos 2a + 4 \cos 3a + \cos 4a = 16 \cos^2 2a \cos^4 \frac{a}{2} \)