Let's derive the equation sin

^{2}x + cos^{2}x=1 from the pythagorean formula.A right triangle with angle x will have a leg that is adjacent to angle x and it will have a leg that is opposite to angle x. There will also be a hypotenuse.

From trigonometry, a right triangle with a given angle x, can defined as follows:

adjacent/hypotenuse=cosx

opposite/hypotenuse=sinx

opposite/hypotenuse=sinx

Following is the pythagorean identity for our right triangle:

adjacent

^{2}+ opposite^{2}=hypotenuse^{2}If we divide the above equation by hypotenuse

^{2}, we have:(adjacent

^{2}+ opposite^{2}=hypotenuse^{2})/hypotenuse^{2}oradjacent

^{2}/hypotenuse^{2}+ opposite^{2}/hypotenuse^{2}=1Substitute in sinx and cosx into the above equation:

cos

^{2}x+sin^{2}x=1Can you derive the other two Pythagorean trigonometric identities?

cot

^{2}x^{ }+ 1=csc^{2}xtan

^{2}x + 1=sec^{2}xHint: These can be derived by using the same right triangle that we used above or these can be derived by dividing cos

^{2}x+sin^{2}x=1 by either sin^{2}x or cos^{2}x.Now you don't need to remember these equations for your trig identity test. You can derive these yourself!