We can simplify the given expression by using the double angle formula for cosine:
cos‌2‌θ=2cos2θ−1This gives us:
2+2‌cos‌8‌θ=2(1+cos‌8‌θ)=4cos24θNow, we can simplify the expression under the radical:
√2+√2+√2+2‌cos‌8‌θ=√2+√2+√4cos24θ=√2+√2+2‌cos‌4‌θ
We can continue this process by repeatedly applying the double angle formula:
√2+√2+2‌cos‌4‌θ=√2+√4cos22θ=√2+2‌cos‌2‌θ
Finally, we have:
√2+2‌cos‌2‌θ=√4cos2θ=2‌cos‌θSince
θ∈[−‌,‌], the cosine function is positive in this interval. Therefore, the final answer is:
√2+√2+√2+2‌cos‌8‌θ=2‌cos‌θ.