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Question : 88 of 180
Marks:
+1 ,
-0
Solution:
We can simplify the given expression by using the double angle formula for cosine:
cos ‌ 2 ‌ θ = 2 cos 2 θ − 1 This gives us:
2 + 2 ‌ cos ‌ 8 ‌ θ = 2 ( 1 + cos ‌ 8 ‌ θ ) = 4 cos 2 4 θ Now, we can simplify the expression under the radical:
√ 2 + √ 2 + √ 2 + 2 ‌ cos ‌ 8 ‌ θ = √ 2 + √ 2 + √ 4 cos 2 4 θ = √ 2 + √ 2 + 2 ‌ cos ‌ 4 ‌ θ
We can continue this process by repeatedly applying the double angle formula:
√ 2 + √ 2 + 2 ‌ cos ‌ 4 ‌ θ = √ 2 + √ 4 cos 2 2 θ = √ 2 + 2 ‌ cos ‌ 2 ‌ θ
Finally, we have:
√ 2 + 2 ‌ cos ‌ 2 ‌ θ = √ 4 cos 2 θ = 2 ‌ cos ‌ θ Since
θ ∈ [ − ‌ , ‌ ] , the cosine function is positive in this interval. Therefore, the final answer is:
√ 2 + √ 2 + √ 2 + 2 ‌ cos ‌ 8 ‌ θ = 2 ‌ cos ‌ θ .
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