CHÚC BẠN HỌC TỐT!!!
Trả lời:
$VT=\cos a+\cos b+\sin(a+b)$
$=2\cos\dfrac{a+b}{2}\cos\dfrac{a-b}{2}+2\sin\dfrac{a+b}{2}\cos\dfrac{a+b}{2}$
$=2\cos\dfrac{a+b}{2}\bigg{(}\cos\dfrac{a-b}{2}+\sin\dfrac{a+b}{2}\bigg{)}$
$=2\cos\dfrac{a+b}{2}\bigg{(}\cos\dfrac{a}{2}\cos\dfrac{b}{2}+\sin\dfrac{a}{2}\sin\dfrac{b}{2}+\sin\dfrac{a}{2}\cos\dfrac{b}{2}+\cos\dfrac{a}{2}\sin\dfrac{b}{2}\bigg{)}$
$VP=4\cos\dfrac{a+b}{2}.\cos\bigg{(}\dfrac{\pi}{4}-\dfrac{a}{2}\bigg{)}.\sin\bigg{(}\dfrac{\pi}{4}+\dfrac{b}{2}\bigg{)}$
$=4\cos\dfrac{a+b}{2}.\bigg{(}\dfrac{\sqrt{2}}{2}.\cos\dfrac{a}{2}+\dfrac{\sqrt{2}}{2}.\sin\dfrac{a}{2}\bigg{)}.\bigg{(}\dfrac{\sqrt{2}}{2}\cos\dfrac{b}{2}+\dfrac{\sqrt{2}}{2}.\sin\dfrac{b}{2}$
$=4\cos\dfrac{a+b}{2}.\dfrac{1}{2}.\bigg{(}\cos\dfrac{a}{2}+\sin\dfrac{a}{2}\bigg{)}.\bigg{(}\cos\dfrac{b}{2}+\sin\dfrac{b}{2}\bigg{)}$
$=2\cos\dfrac{a+b}{2}\bigg{(}\cos\dfrac{a}{2}\cos\dfrac{b}{2}+\sin\dfrac{a}{2}\sin\dfrac{b}{2}+\sin\dfrac{a}{2}\cos\dfrac{b}{2}+\cos\dfrac{a}{2}\sin\dfrac{b}{2}\bigg{)}$
$⇒VT=VP$
$\cos a+\cos b+\sin(a+b)=4\cos\dfrac{a+b}{2}.\cos\bigg{(}\dfrac{\pi}{4}-\dfrac{a}{2}\bigg{)}.\sin\bigg{(}\dfrac{\pi}{4}+\dfrac{b}{2}\bigg{)}$. (Đpcm)
