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sin x + cos x is not equal to 1 for all values of x. This is not a trigonometric identity.
The equation holds for some values of x.
sin x + cos x = 1
=> (sin x + cos x)^2 = 1
=> sin^2x + cos^2x + 2*sin x*cos x = 1
=> 2*sin x*cos x = 0
=> sin 2x = 0
=> `2x = sin^-1 0`
=> x = 0, x = `pi/2 `
The solution of the equation sin x + cos x = 1 is `x = n*2*pi ` and `x = pi/2 + 2*n*pi `
We cannot prove that sin x + cos x = 1 as this is not the case for all values of x.
If we look at the graph of y = sin x + cos x, it is the following:
The values that y takes on lie in the interval `[-sqrt 3, sqrt 3]`
This shows that it is not possible to prove that sin x+cos x=1 as that is not the case.
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