Demonstrate that equation sinx + 2(cos(x/2))^2=1 have roots. find the roots

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to use the following half angle identity, such that:

`cos^2(x/2) = (1 + cos x)/2`

Replacing `(1 + cos x)/2` for the term of equation `cos^2(x/2)` , yields:

`sin x + 2*(1 + cos x)/2 = 1`

Reducing duplicate factors, yields:

`sin x + 1 + cos x = 1`

`sin x + cos x + 1 - 1 = 0 `

`sin x + cos x = 0`

You may divide by `cos x` both sides, such that:

`sin x/cos x + 1 = 0`

Using the trigonometric identity `sin x/cos x = tan x` , yields:

`tan x + 1 = 0`

`tan x = -1 => x = tan^(-1) (-1) + k*pi`

`x = -pi/4 + k*pi`

Hence, checking if the given equation has solutions, the answer is affirmative and the general solution to the equation is `x = -pi/4 + k*pi.`

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