Trig identity hints:

#6: start with the left hand side because it has the sum for the angle. Turn the cotangent into cos/sin. Use the sum of angles formulas for cos(a+b) and sin(a+b). Now divide the numerator and denominator by whatever you need to (sines of the appropriate angles) to turn all of the cosines into cotangents. If you do the algebra correctly, you will get the RHS.

# 8: we're finding a particular formula for cot(x/2) so start with LHS. Turn cot(x/2) into cos(x/2)/sin(x/2). Plug in the half angle formulas for sin and cos, and then multiply by the right version of 1 (*/*) so that the denominator will simplify to the thing you want. Do algebra, and the numerator should work. Finally, look back: will you get the correct sign?

#9: once again, we want a formula for lhs. Follow the same strategy as for #8--turn into sin and cos, plug in formulas. Try to make the denominator be what you want it to be, and simplify.

#10. By now you should know the formula for cos2u pretty well, so start with RHS, and try to make it look like one of the cos2u formulas.