y=1\3logtan(cosx+sinx\cosx-sinx)
To find the value of y given the equation y = (1/3) log tan[(cosx + sinx)/(cosx - sinx)], we can start by simplifying the expression inside the tangent function.
Let's begin by looking at the trigonometric identity: tan A = (sin A) / (cos A), where A represents an angle.
In our case, A = [(cosx + sinx) / (cosx - sinx)].
Now, let's simplify A further:
A = (cosx + sinx) / (cosx - sinx)
To simplify the expression, we can multiply the numerator and the denominator by the conjugate of the denominator, which is (cosx + sinx).
A = [(cosx + sinx) / (cosx - sinx)] * [(cosx + sinx) / (cosx + sinx)]
Expanding the numerator and denominator:
A = [(cosx + sinx)(cosx + sinx)] / [(cosx - sinx)(cosx + sinx)]
Simplifying further:
A = [(cosx + sinx)^2] / [cos^2(x) - sin^2(x)]
Now, let's use the identity sin^2(x) + cos^2(x) = 1 to further simplify the expression:
A = [(cosx + sinx)^2] / [(1 - 2sin^2(x))]
Expanding the numerator:
A = [cos^2(x) + 2cosxsinx + sin^2(x)] / [1 - 2sin^2(x)]
Simplifying:
A = [1 + 2cosxsinx] / [1 - 2sin^2(x)]
Now, substitute A back into the original equation:
y = (1/3) log tan A
y = (1/3) log tan {[1 + 2cosxsinx] / [1 - 2sin^2(x)]}
Note: Since we can't explicitly solve for y in terms of x using simple algebraic manipulations, we can rely on numerical methods or graphing tools to find the values of y for different values of x.