It's your turn on trigonometric identities

Do your best in trying to solve the following problems. In any case some of them are solved in the video and all of them are solved in the pdf file below.

Exercise 1.

Compute the following values [1) sin(pi/4+pi/3) text{ and } sin(pi/4-pi/3);] [2) cos(pi/6+2pi/3) text{ and } cos(pi/6-2pi/3).]

Exercise 2.

Simplify the following expressions [1) dfrac{sin^4 x-cos^4 x}{sin^2 x-cos^2 x};] [2) dfrac{1}{cos x(1+tan^2x)}.]

Exercise 3.

Prove the following identities [1) cos^4x-sin^4x=cos(2x);] [2) dfrac{tan x}{1+tan^2x}=sin xcos x;] [3) sin xcos xtan x=1-cos^2x;] [4) dfrac{sin x}{1-cos x}+dfrac{1-cos x}{sin x}=dfrac 2{sin x};] [5) dfrac{sin x-cos x}{sin x+cos x}=-dfrac{cos 2x}{1+sin 2x}.]