How can we prove a given identity?

[mathslover]

How can we proceed to prove the following identity ?

 

[Rogerio]

The original expression :
$$\sqrt{22+2\sqrt{5}\ }\ +\ \sqrt{5}\ = \ \sqrt{11+2\sqrt{29}\ }\ +\ \sqrt{16-2\sqrt{29}+2\sqrt{55-10\sqrt{29}\ }\ }$$

First, note that 16=11+5 .

So, we can rewrite the last part :

$$\sqrt{22+2\sqrt{5}\ }\ +\ \sqrt{5}\ = \ \sqrt{11+2\sqrt{29}\ }\ +\ \sqrt{ ( \ \sqrt{11-2\sqrt{29}\ }\ +\ \sqrt{5}\ ) ^2\ }$$

Or:

$$\sqrt{22+2\sqrt{5}\ }\ +\ \sqrt{5}\ = \ \sqrt{11+2\sqrt{29}\ }\ +\ (\ \sqrt{11-2\sqrt{29}\ }\ +\ \sqrt{5}\ )$$

Thats the same as:

$$\sqrt{22+2\sqrt{5}\ }\ = \ \sqrt{11+2\sqrt{29}\ }\ +\ \sqrt{11-2\sqrt{29}\ }$$

Squaring both sides, we get:

$$22+2\sqrt{5}\ = 22\ +\ 2\sqrt{11^2\ -\ (2\sqrt{29})^2\ }$$

Or:

$$22+2\sqrt{5}\ = 22\ +\ 2\sqrt{121\ -\ 4*29\ }$$

Which is true.

 

[tacman]

Proving an identity often involves breaking it down into smaller, more manageable parts and using known mathematical principles and properties to show that both sides of the equation are equal. It may also involve using algebraic manipulations or substitution to simplify or transform the expression.

One approach to proving an identity is to start with one side of the equation and manipulate it until it matches the other side. For example, if the identity involves trigonometric functions, you may need to use trigonometric identities or the unit circle to simplify the expression.

Another approach is to work backwards from the desired result and find a series of steps that will lead to the given identity. This may involve using the distributive property, combining like terms, or factoring.

In some cases, it may be helpful to use a proof by induction, which involves showing that the identity holds for a base case and then assuming it holds for a general case and proving that it holds for the next case.

Ultimately, the key to proving an identity is to carefully analyze the given expression and use logical steps to show that it is equivalent to the desired result. It may require creativity, patience, and a thorough understanding of mathematical concepts.