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**cuteisa89** 1. Simplify the expression sinx (cscx - sin x)

sinx(cscx - sinx) = sinx(1/sinx - sinx)

.......................= sinx/sinx - sinx*sinx

.......................= 1 - sin^2x

.......................= cos^2x

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2. rewrite the expression cos^2 / (1 + sin x) so that there is no fraction

cos^2x/(1 + sinx) = (1 - sin^2x)/(1 + sinx) ...the top is the difference of two squares

.........................= [(1 + sinx)(1 - sinx)]/(1 + sinx)

.........................= 1 - sinx

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3. use the half-angle formula to simplify the expression sqrt ((1+cos(4x))/2)

by the half angle formula:

cos(x/2) = sqrt[(1 + cosx)/2]

if we replace x with 4x we get:

cos(2x) = sqrt[(1 + cos(4x))/2]

so cos(2x) is the simplified expression

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4. Use the following to find the exact value of sin(x/2). cosx = -4/5, x is in quadrant III

sin(x/2) = - sqrt[(1 - cosx)/2] ..........sine is negative in quad III

we are given that cosx = -4/5

=> sin(x/2) = -sqrt[(1 - (-4/5))/2]

................= -sqrt[(9/5)/2]

................= -sqrt(9/10)

................= -3/sqrt(10)

................= -3sqrt(10)/10

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5. Evaluate the expression sin^2(355pi/ 113) + cos^2 (355pi/113)

sin^2(355pi/ 113) + cos^2 (355pi/113) = 1

since sin^2x + cos^2x = 1 for all x

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6. verify the identity cos x/(1 + sinx) + cosx/(1-sinx) = 2 secx

Tips for tackling such problems:

1) start on the most complicated side, it gives you more options to change things

2) it is USUALLY better to change everything to sines and cosines. these are the ones students are more familiar with, so it's easier to see connections

3) always consider working on both sides one at a time to bring them to the same thing. sometimes working on one side only to get the answer is too hard or even impossible in some sense

see below for solution