Question

 

A=(cos a +2cos2a + cos 3a)/(sin a + sin2a + sin3a)

Answer 1

\(A=\dfrac{cosa+2\cdot cos2a+cos3a}{sina+sin2a+sin3a}\)

\(=\dfrac{\left(cosa+cos3a\right)+2\cdot cos2a}{\left(sina+sin3a\right)+sin2a}\)

\(=\dfrac{2\cdot cos\left(\dfrac{3a+a}{2}\right)\cdot cos\left(\dfrac{3a-a}{2}\right)+2\cdot cos2a}{2\cdot cos\left(\dfrac{3a-a}{2}\right)\cdot sin\left(\dfrac{3a+a}{2}\right)+sin2a}\)

\(=\dfrac{2\cdot cos2a\cdot cosa+2\cdot cos2a}{2\cdot cosa\cdot sin2a+sin2a}=\dfrac{2\cdot cos2a\left(cosa+1\right)}{sin2a\left(2\cdot cosa+1\right)}\)

\(=\dfrac{2\cdot cosa+2}{2\cdot cosa+1}\cdot cot2a\)