Trigonometric identities are formulas that involve Trigonometric functions. Trigonometry is a branch of mathematics that deals with the study of relationships between lengths and angles of triangles. The early applications of the trigonometric functions were to surveying, navigation, and engineering. These functions also play an important role in the study of all sorts of vibratory phenomenasound, light, electricity, etc.
Trigonometric Ratios
Let us take a right triangle ABC as shown below.
The trigonometric ratios of the angle A in right triangle ABC are defined as follows :
Note that the ratios cosec A, sec A and cot A are respectively, the reciprocals of the ratios sin A, cos A and tan A. So, the trigonometric ratios of an acute angle in a right triangle express the relationship between the angle and the length of its sides. These are trigonometric ratios for with respect to acute angle A. However, these ratios can be extended to other angles also.
Trigonometric ratios of some specific angles
not defined  
not defined  2  
2  not defined  
not defined  0 
Remark : From the table above you can observe that as ∠ A increases from 0° to 90°, sin A increases from 0 to 1 and cos A decreases from 1 to 0.
Trigonometric Ratios of Complementary Angles
Trigonometric Identities
An equation involving trigonometric ratios of an angle is called a trigonometric identity, if it is true for all values of the angle(s) involved. Some basic trigonometric identities are :
(i)
(ii)
(iii)
These are the trigonometric ratios of acute angles as the ratio of the sides of a right angled triangle. Further the concept of trigonometric ratios is extended to the trigonometric ratios to any angle in terms of radian measure and study them as trigonometric functions.
Trigonometric functions
Consider a unit circle with centre 0. In right angle triangle OMP , and . Since
triangle OMP is a right triangle , we have
Now, if we take one complete revolution from the point P, we again come back to
same point P. Thus, we also observe that if increases (or decreases) by any integral
multiple of , the values of sine and cosine functions do not change. Thus,
and
Further
, if i.e. when is an integral multiple of 
and
, if , i.e. vanishes when is an odd multiple of 
Thus and
Other trigonometric functions are
, , where is any integer
, , where is any integer
, , where is any integer
, , where is any integer
Therefore we have the following identities

Trigonometric ratios of higher angles:
Degree  
Radian  
sin  0  1  0  1  0  
cos  1  0  1  0  1  
tan  0  1  not defined  0  not defined  0  
cot  not defined  1  0  not defined  0  not defined  
sec  1  2  not defined  1  not defined  1  
cosec  not defined  2  1  not defined  1  not defined 
Sign of trigonometric functions in different quadrants:
I  II  III  IV  
+  +  –  –  
+  –  –  +  
+  –  +  –  
+  +  –  –  
+  –  –  +  
+  –  +  – 
Note: The signs (positive or negative) of the trigonometric ratios in the four quadrants can be remembered as
shown below.
I  II  III  IV 
All  Silver  Tea  Cups 
 In quadrant I, all are positive.
 In quadrant II, only sine (and its reciprocal cosec) is positive.
 In quadrant III, only tangent (and its reciprocal cot) are positive.
 In quadrant IV, only cosine (and its reciprocal sec) is positive.
Trigonometric Functions of Sum and Difference of Two Angles
 If none of the angles and is an odd multiple of , then
 If none of the angles and is an odd multiple of , then
 If none of the angles and is a multiple of , then
 If none of the angles and is a multiple of , then
 , if , n is an integer
 where n is an integer