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e.g., f(x) = x3, f(-x) = (-x)3 = -x3 = -f(x)
f(x) = cos x is even for f(-x) = cos (-x) = cos q = f(x)f(x) = x cos x is odd for f(-x) = (-x) cos (-x) = -x cos x = -f(x)
f(x) = sin x is odd whereas f(x) = x sin x is even.Theorem 6
Statement:
For all real numbers x and y

Let x and y be any two real numbers and let P(x)and Q(y) be the corresponding trigonometric points on the unit circle. In the above figures we have taken x and y so that 
Now choose a point R on the unit circle so that arc AR has a measure of (x-y) units. Then the trigonometric point with respect to (x-y) is R(x-y) and the corresponding coordinates of R are (cos(x-y),sin(x-y)). The arc length of AR is the same as arc PQ and hence the chord lengths of PQ and AR are same.
i) |AR| = distance from R to A.










Theorem 7
Statement:
For all values of 


Proof:








Theorem 8
For all real values of x.


Proof:







Theorem 9
For real values of x

Proof:




Theorem 10


Proof:




Theorem 11



Proof:












= 

Theorem 12



Proof:







Theorem 13

Proof:






Theorem 14



Proof:
For all A and B we have
Let A + B = x, A - B = y, then
Adding (b) and (a), we get




Subtracting (d) from (c), we get

Theorem 15

Proof:







