. Trigonometric Identities For all real values of q , we have the following three identities. (i) sin = + cos (ii) +tan = sec (iii) cot q cosec These identities are termed as three fundamental identities of trigonometry. We will now prove them as follows.
Picture Identity Proof P M O Fig. . In the right angled ∆ OMP , we have OM OP PM OP cos , q ... ( ) By Pythagoras theorem MP OM OP ...
( ) Dividing each term on both sides of ( ) by OP , ( OP ≠ ) we get, MP OP OM OP OP OP Þ MP OP OM OP OP OP + = From ( ) , (sin ) (cos ) Hence sin In the right angled ∆ OMP, we have MP OM OP OM tan , sec q ... ( ) Trigonometry tan sec From ( ) , MP OM OP Dividing each term on both sides of ( ) by OM , ( OM ¹ ) we get, MP OM OM OM OP OM Þ MP OM OM OM OP OM + = From ( ) , (tan ) (sec ) Hence tan sec q . + cot q = cosec q In the right angled ∆ OMP, we have OM MP OP MP cot , q cosec ... ( ) From ( ) , MP OM OP Dividing each term on both sides of ( ) by MP , ( MP ¹ ) we get, MP MP OM MP OP MP Þ MP MP OM MP OP MP + = From ( ) , (cot ) cosec Hence, cot q cosec These identities can also be rewritten as follows.
Identity Equal forms (or) cos tan sec tan sec − (or) sec tan cot q cosec cot cosec (or) cosec cot