Identities Test Correction and Unit Reflection

I seriously struggled with this unit. I never quite got it. I can do things with formulas much better in math than proofs.

5) tan^2x/secx-1=secx+1 - Prove

1) sin^x/cos^2x/                                               1) Put the left side of the equation in terms of sine and cosine
1/cosx-1

2) sin^2x/cos^2x/                                                                                       2) Change 1 to (cosx/cosx)
1/cosx-(cosx/cosx)

3)sin^2x/cos^2x/                                 3) Combine the numerators because there is a common denominator
1-cosx/cosx

4) sin^2x/cox^2x*cosx/1-cosx                                                                 4) Multiply by the reciprocal

5) sin^2x/                                                                                                    5) Multiplication done out
cosx(1-cosx)

6) 1-cos^2x/                                                         6) Change sin^2x to 1-cos^2x using Pythagorean identity 
cosx(1-cosx)

7) (1+cosx)(1-cosx)/                                                                                      7) Factor out 1-cos^2x
(cosx)(1-cosx)

8) 1+cosx/                                                                                                           8)Cancel out 1-cosx
cosx

9) 1/cosx+cosx/cosx                                                                                    9) Separate the numerator

secx+1





8) 1/cotx+tanx=sinxcosx   -Prove

1) 1/((cosx/sinx)+(sinx/cosx))                                                     1) Put in terms of sine and cosine

2) 1/ ((cos^2x/(sinxcosx))+sin^2x/(sinxcosx))                           2) Make common denominators

3) 1/ (cos^2x+sin^2x)/(sinxcosx)                                              3) Combine numerators

4) 1/1/(sinxcosx)                                                                       4) Simplify

sinxcosx

Vectors Test Corrections and Unit Reflection

I understood this unit fairly well. The component method made more sense to me than the law of cosines method, though I could use the law of cosines when I had to. Sometimes with the component method I wasn't sure which angle to solve for. On the test, the problem I struggled with (number 2) was hard for me because I couldn't figure out how to draw the picture. The pictures are important for me.

2a) Use simple trig to calculate the speed of the current. You know the angle is 34 degrees and the hypotenuse is 4 km/hr. You need to calculate the opposite side, so you use sine.
sin34=x/4
x=2.24 km/hr.

2c) Here, you know that the adjacent side to the angle you need is 4 km/hr and the opposite side is 2.24. In order to find the angle, you can use tangent.
tan2.24/4=
29.25 degrees