Anonymous
Anonymous asked in Science & MathematicsMathematics · 7 months ago

# If cos x +cos y=1, sec x+sec y=4, where 0°<x<180°, find value of x+y.?

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• Mewtwo
Lv 5
7 months ago

Using the second equation:

4 = sec x + sec y

= 1 / cos x + 1 / cos y

= (cos y + cos x) / (cos x cos y)

= 1 / (cos x cos y), or

cos x cos y = 1/4.

Using the first equation, cos y = 1 - cos x. Plugging this into the equivalent form of the second equation that we found, then,

cos x(1 - cos x) = 1/4, or

4 cos x(1 - cos x) = 1.

Thus,

4 cos x - 4 cos^2 x = 1

4 cos^2 x - 4 cos x + 1 = 0

Factoring this gives

(2 cos x - 1)^2 = 0.

Thus,

2 cos x - 1 = 0

cos x = 1/2.

Since 0° < x < 180°, we verify then that x = 60° (Use your unit circle or a calculator).

Thus,

cos y = 1 - cos x

= 1 - cos 60°

= 1 - 1/2

= 1/2.

So, y = 60° as well. Therefore, x + y = 60° + 60° = 120°.