Let's derive the equation sin2x + cos2x=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:
Following is the pythagorean identity for our right triangle:
adjacent2 + opposite2=hypotenuse2
If we divide the above equation by hypotenuse2, we have:
(adjacent2 + opposite2=hypotenuse2)/hypotenuse2 or
adjacent2/hypotenuse2 + opposite2/hypotenuse2=1
Substitute in sinx and cosx into the above equation:
Can you derive the other two Pythagorean trigonometric identities?
cot2x + 1=csc2x
tan2x + 1=sec2x
Hint: These can be derived by using the same right triangle that we used above or these can be derived by dividing cos2x+sin2x=1 by either sin2x or cos2x.
Now you don't need to remember these equations for your trig identity test. You can derive these yourself!