Home Light Version Normal Version Search | BBS | Guest Book | Site Map | Help | Contact us | About us

### [Index]

Introduction
Essence
Memorizing
 How to memorize Basic Formulas Sum and Difference Formulas Double & Triple Angle Formulas Half Angle Formulas Product to Sum Formulas Sum to Product Formulas All Formulas
Understanding
 Page 1 Page 2 Page 3 Page 4 Page 5
Summarizing
 Page 1 Page 2
Examples
 Examples Example 1 Example 2 Example 3 Example 4 Example 5 Example 6 Example 7 Example 8 General ideas
Exercises
Final thoughts
Page 1 | Page 2 | Page 3 | Page 4 | Page 5 | Proof(s)

### Proofs of example 8:

Suppose that a, b, c, , and are real numbers, with the following conditions: , , k is any integer.

 (1) (2)

Prove that .

First proof:

 (1) + (2) (3)

that is,

 Dividing by 2, (4)
 Squaring the both sides, (5)

Claim

 (6)
 (1) - (2) (7)

 (8)

From , we have:
From equation (8), we have:

 (9)

From (6), we have:

 (10)

Squaring (9) and adding it to (10), we have:

From equations (5) and (6), we have:

Since ,

Second proof:

 From (1) (3) From (2) (4)

where satisfies , or (if a = 0).

 (3) + (4) that is,

Since ,

 Squaring the both sides, (5)

 Claim: (6)

(3) - (4)

Since ,

 that is, (7)

Since ,

so

Since , .  Hence (6) holds. From equations (5) and (6) we have: