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Bài 4. Chứng minh các đẳng thức
a) \( \frac{cos(a-b)}{cos(a+b)}=\frac{cotacotb+1}{cotacotb-1}\)
b) \(\sin(a + b)\sin(a – b) = \sin^2a – \sin^2b = \cos^2b – \cos^2a\)
c) \(\cos(a + b)\cos(a – b) = \cos^2a – \sin^2b = \cos^2b – \sin^2a\)
a) \(VT = {{\cos a\cos b+\sin a\sin b}\over{\cos a\cos b-\sin a\sin b}}=\frac{\frac{\cos a\cos b}{\sin a\sin b}+1}{\frac{\cos a\cos b}{\sin a\sin b}-1}=\frac{\cot a\cot b+1}{\cot a\cot b-1}\)
b) \(VT = [\sin a\cos b + \cos a\sin b][\sin a\cos b – \cos a\sin a]\)
\(= (\sin a\cos b)^2– (\cos a\sin b)^2= \sin^2 a(1 – \sin^2 b) – (1 – \sin^2 a)\sin^2 b\)
\(= \sin^2a – \sin^2b = \cos^2b( 1– \cos^2a) – \cos^2 a(1 – \cos^2 b) = \cos^2 b – \cos^2 a\)
c) \(VT = (\cos a\cos b – \sin a\sin b)(\cos a\cos b + \sin a\sin b)\)
\(= (\cos a\cos b)^2 – (\sin a\sin b)^2\)
\(= \cos^2 a(1 – \sin^2 b) – (1 – \cos^2 a)\sin^2 b = \cos^2 a – \sin^2 b\)
\(= \cos^2 b(1 – \sin^2 a) – (1 – \cos^2 b)\sin^2 a = \cos^2 b – \sin^2 a\)