Question

simplify 2/√2+√2+√2+cos 4x​

Answer 1

Answer:  

Solution:

2/[√2 + {√2 + √(2 + 2 cos 4x)}]….(i)

2 + 2 cos 4x = 2{1 + cos 2(2x)} = 2{2 cos2(2x)} = 4 cos2(2x)

√[4 cos2(2x)] = 2 cos 2x

Now,

√2 + √(2 + 2 cos 4x) = √(2 + 2 cos 2x) = √2(1 + cos 2x) = √2(2 cos2x) = 2 cos x

And

√2 + {√2 + √(2 + 2 cos 4x)} = √(2 + 2 cos x) = √2(1 + cos x) = √2[2 cos2(x/2)] = 2 cos(x/2)….(ii)

From (i) and (ii),

2/[√2 + {√2 + √(2 + 2 cos 4x)}] = 2/[2 cos(x/2)] = sec x/2