Question

Solve: $\cos A+\sqrt{2}\cos ({135}^{\circ }+A)+\sqrt{2}\cos ({135}^{\circ }-A)$

Answer 1

$cosA+\sqrt{2}cos(135+A)+\sqrt{2}cos(135-A)$

we know

cos(a+b)+cos(a-b) = 2cosa cosb.

$\Rightarrow cosA+\sqrt{2}\left[cos\right(135+A)+cos(135-A\left)\right]$

$\Rightarrow cosA+2\sqrt{2}cos135.cosA.$

$cosA[1+2\sqrt{2}cos135]$

$cos135=\frac{-1}{\sqrt{2}}$

$\Rightarrow cosA[1+2\sqrt{2}\times (\frac{-1}{\sqrt{2}}\left)\right]$

cosA [1-2] = - cosA.