# How do you find the limit of #(1-tanx)/(sinx-cosx)# as #x->pi/4#?

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To find the limit of (1 - tanx)/(sinx - cosx) as x approaches pi/4, we can use algebraic manipulation and trigonometric identities.

First, let's simplify the expression by multiplying the numerator and denominator by the conjugate of the denominator, which is (sinx + cosx):

[(1 - tanx)/(sinx - cosx)] * [(sinx + cosx)/(sinx + cosx)]

Expanding this expression, we get:

[(sinx + cosx - tanx*sinx - tanx*cosx)/(sinx - cosx)]

Next, we can simplify further by factoring out sinx from the numerator:

[sinx * (1 - tanx) + cosx * (1 - tanx)] / (sinx - cosx)

Now, we can cancel out the common factor of (1 - tanx):

[sinx + cosx] / (sinx - cosx)

Using the trigonometric identity sinx = cos(pi/2 - x), we can rewrite the expression as:

[cos(pi/2 - x) + cosx] / [cos(pi/2 - x) - cosx]

Applying the trigonometric identity cos(A) - cos(B) = -2sin((A + B)/2)sin((A - B)/2), we can simplify further:

[-2sin((pi/4 + x)/2)sin((pi/4 - x)/2)] / [-2sin((pi/4 + x)/2)sin((pi/4 - x)/2)]

The sin((pi/4 + x)/2) and sin((pi/4 - x)/2) terms cancel out, leaving us with:

-1

Therefore, the limit of (1 - tanx)/(sinx - cosx) as x approaches pi/4 is -1.

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