An identity is a mathematical statement which equates any two mathematical expressions. In algebra we have the following identities which are mostly used in factoring the expressions.
(a+b)2 = a2 + 2ab + b2
(a-b)2 = a2 – 2ab + b2
Similarly, we have identities even in Trigonometry which are mostly used in solving trigonometric equations.
Let us assume any point (x,y) be a point in the first quadrant on an unit circle.
Then by the definitions of trigonometric ratios, we have
Sin t = y; Cost = x; Tan t = y/x; Cosec t = 1/y; Sec t = 1/x; Cot t = x/y
From these definitions itself, we can derive the following elementary trigonometric identities.
Sin t = 1/ Cosec t;
Cos t = 1/ Sec t;
Tan t = 1/Cot t;
Also, Tant = Sint/Cost; Cott = Cost/Sint
Beyond these elementary trigonometric identities, we have some other identities known as Pythagorean identities.
By Pythagoras theorem, we know that, in a right angled triangle the square of the hypotenuseis equal to sum of the squares of the other two sides.
By applying this theorem to the above figure we get.
x2 + y2 = 1
Cos2t + Sin2t = 1;
By dividing this equation by Cos2t, we get
Cos2t / Cos2t + Sin2t/ Cos2t = 1/ Cos2t
1 + Tan2t= Sec2t;
Again by dividing the same equation by Sin2t, we get
Cos2t/ Sin2t + Sin2t/ Sin2t = 1/ Sin2t
Cot2t + 1 = Cosec2t;
Thus we are derived the Pythagorean identities which are listed as follows.
Cos2t + Sin2t = 1;
1 + Tan2t= Sec2t;
Cot2t + 1 = Cosec2t;
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