If e^(cos² x + cos⁴ x + cos⁶ x +....) = logₑ 2 satisfies the equation t² - 9t + 8 = 0, then the value of (2 sin x)/(sin x + √3 cos x) (0

Question

If e^(cos² x + cos⁴ x + cos⁶ x +....) = logₑ 2 satisfies the equation t² - 9t + 8 = 0, then the value of (2 sin x)/(sin x + √3 cos x) (0 < x < π/2) is

Accepted Answer

1/2. The equation e^(cos² x + cos⁴ x + cos⁶ x +...) simplifies to a geometric series, which converges to a specific value. The roots of the quadratic equation t² - 9t + 8 = 0 are t = 1 and t = 8, and substituting these values leads to the correct trigonometric ratio, yielding the final answer of 1/2.

If e^(cos² x + cos⁴ x + cos⁶ x +....) = logₑ 2 satisfies the equation t² - 9t + 8 = 0, then the value of (2 sin x)/(sin x + √3 cos x) (0 < x < π/2) is

Solution

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