An identity is a relation which is always true. To emphasise this the symbol ‘ ’ is often used rather than ‘ ’. For example, (always true) but (only true for ).
Using the exact values, evaluate
[Note that means means ]
- Choose a non-integer value for and use a calculator to evaluate .
- The answer should be whatever value you choose.
For any value of
One way of proving the result in Key Point 12 is to use the definitions of and obtained from the circle of unit radius. Refer back to Figure 22 on page 23.
Recall that , . By Pythagoras’ theorem
We have demonstrated the result (5) using an angle in the first quadrant but the result is true for any i.e. it is indeed an identity.
By dividing the identity by
obtain two further identities.
[Hint: Recall the definitions of .]
Key Point 13 introduces two further important identities.
Note carefully the addition sign in (6) but the subtraction sign in (7).
Further identities can readily be obtained from (6) and (7).
Dividing (6) by (7) we obtain
Dividing every term by we obtain
Replacing by in (6) and (7) and remembering that we find
Using the identities and
obtain an expansion for :
Dividing every term by gives
The following identities are derived from those in Key Point 13.