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Easy derivation of pythagorean trigonometric identities

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:
adjacent/hypotenuse=cosx
opposite/hypotenuse=sinx
 
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:
 
cos2x+sin2x=1
 
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!
 
 
 
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