Question

which expression equals cos 2x for all x€R

Answer 1

y=cosx+cos2xy=cos⁡x+cos⁡2x

⟹y=cosx+2cos2x−1⟹y=cos⁡x+2cos2⁡x−1

⟹y=2(cos2x+12cosx)−1⟹y=2(cos2⁡x+12cos⁡x)−1

⟹y=2(cos2x+2cosx×14+116−116)−1⟹y=2(cos2⁡x+2cos⁡x×14+116−116)−1

⟹y=2(cosx+14)2−18−1⟹y=2(cos⁡x+14)2−18−1

⟹y=2(cosx+14)2−98⟹y=2(cos⁡x+14)2−98

Minimum value =−98=−98 [concave up parabola]