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

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

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

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

cos^2x/(1 + sinx) = (1 - sin^2x)/(1 + sinx) ...the top is the difference of two squares2. rewrite the expression cos^2 / (1 + sin x) so that there is no fraction

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

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

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

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

sin(x/2) = - sqrt[(1 - cosx)/2] ..........sine is negative in quad III4. Use the following to find the exact value of sin(x/2). cosx = -4/5, x is in quadrant 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

sin^2(355pi/ 113) + cos^2 (355pi/113) = 15. Evaluate the expression sin^2(355pi/ 113) + cos^2 (355pi/113)

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

Tips for tackling such problems:6. verify the identity cos x/(1 + sinx) + cosx/(1-sinx) = 2 secx

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