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Bài 3. Rút gọn các biểu thức
a) \(\sin(a + b) + \sin(\frac{\pi}{2}- a)\sin(-b)\).
b) \(cos(\frac{\pi }{4} + a)\cos( \frac{\pi}{4} – a) + \frac{1 }{2} \sin^2a\)
c) \(\cos( \frac{\pi}{2} – a)\sin( \frac{\pi}{2} – b) – \sin(a – b)\)
a) \(\sin(a + b) + \sin( \frac{\pi }{2} – a)\sin(-b) = \sin a\cos b + \cos a\sin b – \cos a\sin b = \sin a\cos b\)
b) \(\cos( \frac{\pi }{4} + a)\cos(\frac{\pi }{4}- a) + \frac{1 }{2}\sin^2a\)
\( =\frac{1 }{2}\cos\left [ \frac{\pi }{4}+a+\frac{\pi}{4} -a\right ]+\frac{1}{2}\cos\left [ \left ( \frac{\pi }{4} +a\right ) -\left ( \frac{\pi}{4}-a \right )\right ]+\frac{1}{2}\left ( \frac{1-\cos 2a}{2} \right )\)
\( =\frac{1}{2}\cos 2a + \frac{1}{4}(1 – \cos 2a) = \frac{1+\cos 2a}{4 }= \frac{1 }{2}\cos^2 a\)
c) \(\cos( \frac{\pi}{2} – a)\sin( \frac{\pi}{2} – b) – \sin(a – b) = \sin a\cos b – \sin a\cos b + \sin b\cos a\)
\(= \sin b\cos a\)