Trigonometric Identities | Theorems Based on Trigonometric Identities

Trigonometric Identies are some identies used in Trigonometry in order to make the calculations easier.
Trigonometry is a word consisting of three Greek words " Tri" means three, "Gon" means side, and "Metron" means measure. Thus, trigonometry is a study related to the measures of sides and angles of a triangle. Trigonometry is mainly used by captains of ships to find the direction and distance of islands and light houses from sea. Trigonometry is also used in astronomy, geography and engineering.
Trigonometric Ratios
In any right-angled triangle ABC,
let  angle B = 90 o  and angle C = T.                                                      

Line segment AC is the hypotenuse.
With reference to angle C, we can say that,
Line segment AB is the opposite side of Line segment BC is the adjacent side of Therefore, trigonometric ratios are given as,


Trigonometric Identities
Basic trigonometric identities are:
sin^2 T + cos^2 T = 1.
tan^2 T + 1 = sec^2 T
1 + cot 2T  = cosec^2 T
Theorems Based on Trigonometric Identities

Theorem 1: sin^2 T + cos^2 T =1
In right-angled triangle ABC, let angle< B = 90, angle< C = T.
Let AB = a, BC = b, and AC = c
By Pythagorean theorem we can say,
(hypotenuse)^2 = ( side)^2 + (side)^2
From figure we can say,
(AC)^2 =  (AB)^2 + (BC)^2
c^2 =  a^2 + b^2
divide throughout by c^2, we get,
(c^2 ) / c^2 =  ( a^2 + b^2 ) / c^2
1  =  a^2 / c^2 + b^2 / c^2
=  ( AB )^2 / ( AC)^2 + ( BC)^2 / (AC)^2
=   (AB / AC)^2 + ( BC / AC)^2
=   ( sin T )^2 +  ( cos T )^2
Therefore,
1 = sin^2 T + cos^2 T

Theorem 2: tan2 T + 1 = sec2 T
We have sin^2 T + cos^2 T = 1
Divide on both sides by cos^2 T,
( sin^2 T + cos^2 T ) / cos^2 T  =  1 / cos^2 T
(sin^2 T / cos^2 T) + (cos^2 T / cos^2 T)  =  1 / cos^2 T
By using trigonometric ratios,
sin T/ cos T  =  tan T
1 / cos T  =  sec T
substitute the values we get,
( sin T / cos T )^2 + 1  =  ( 1 / cos T)^2
(tan T)^2 + 1  =  ( sec T )^2
tan^2 T + 1  =  sec^2 T
Theorem 3: 1 + cot2 T  = cosec^2 T
We have sin^2 T + cos^2 T = 1
Divide on both sides by sin^2 T,
( sin^2 T + cos^2 T ) / sin^2 T  =  1 / sin^2 T
(sin^2 T / sin^2 T) + (cos^2 T / sin^2 T)  =  1 / sin^2 T
By using trigonometric ratios,
cos T/ sin T  =  cot T
1 / sin T  =  cosec T
substitute the values we get,
1 +  ( cos T / sin T )^2   =  ( 1 / sin T)^2
1 +  (cot T)^2  =  ( cosec T )^2
1 +  cot 2T  =  cosec^2 T