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Question

Accepted Answer

Given: (tan(x) / sec(x)) = 1/2

Step 1: Express tan(x) and sec(x) in terms of sin(x) and cos(x):
- tan(x) = sin(x) / cos(x)
- sec(x) = 1 / cos(x)

Step 2: Substitute these into the given equation:
(tan(x) / sec(x)) = (sin(x)/cos(x)) / (1/cos(x)) = (sin(x)/cos(x)) * (cos(x)/1) = sin(x)

So, sin(x) = 1/2.

Step 3: We need to find (sin(x) + cos(x))^2.
Recall the identity:
(sin(x) + cos(x))^2 = sin^2(x) + cos^2(x) + 2 sin(x) cos(x) = 1 + 2 sin(x) cos(x)

Step 4: Find cos(x) using sin(x) = 1/2.
Since sin(x) = 1/2, x could be 30° or π/6 radians (considering principal values).

cos(30°) = √3 / 2

Step 5: Calculate (sin(x) + cos(x))^2:
= 1 + 2 * (1/2) * (√3 / 2) = 1 + 2 * (√3 / 4) = 1 + (√3 / 2) = (2/2) + (√3 / 2) = (2 + √3) / 2

Therefore, the value is (2 + √3) / 2.

Answer

Answer

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