a) Áp dụng công thức \(\cos (a+b)\) với VT sau đsó chia cả tử và mẫu cho \(\sin a \sin b\) ta được:
\(VT = {{\cos a\cos b+\sin a\sin b}\over{\cos a\cos b-\sin a\sin b}}\\=\frac{\frac{\cos a\cos b}{\sin a\sin b}+1}{\frac{\cos a\cos b}{\sin a\sin b}-1}=\frac{\cot a\cot b+1}{\cot a\cot b-1}.\)
\(b) \, \, VT = [\sin a\cos b + \cos a\sin b].\)\([\sin a\cos b - \cos a\sin a] \\ = (\sin a\cos b)^2– (\cos a\sin b)^2 \\ = \sin^2 a(1 – \sin^2 b) – (1 – \sin^2 a)\sin^2 b\\= \sin^2a – \sin^2b \, \, (đpcm) \\ = \cos^2b( 1– \cos^2a) – \cos^2 a(1 – \cos^2 b) \\ = \cos^2 b – \cos^2 a \, \, (đpcm). \)
\(c) \, \, VT = (\cos a\cos b - \sin a\sin b).\)\((\cos a\cos b + \sin a\sin b) \\ = (\cos a\cos b)^2 – (\sin a\sin b)^2\\ = \cos^2 a(1 – \sin^2 b) – (1 – \cos^2 a)\sin^2 b \\ = \cos^2 a – \sin^2 b \, \, (đpcm) \\ = \cos^2 b(1 – \sin^2 a) – (1 – \cos^2 b)\sin^2 a\\ = \cos^2 b – \sin^2 a\, \, (đpcm) \)
