# Proving Identities

Verifying/Proving Trig Identities (Part 1)

Tip: This isn't the place to ask a question because the teacher can't reply.

1 of 7 videos by turksvids

## Key Questions

• Identity 1:

Proof :
For a triangle where a is perpendicular, b is base and h is hypotenuse.

Identity 2 :

Proof:
Divide identity 1 by ${\cos}^{2} \left(x\right)$ .

Identity 3:

Proof:
Divide identity 1 by ${\sin}^{2} \left(x\right)$ .

• If you want to prove a statement using identities you need to rewrite the problem with equivalent forms of identities and combine them to lead to your statement.

For example:
Prove tan(x) + sec(x) = $\frac{\sin \left(x\right) + 1}{\cos} \left(x\right)$

Use the Pythagorean identity of secant and tangent to rewrite the problem as:

$\sin \frac{x}{\cos} \left(x\right) + \frac{1}{\cos} \left(x\right) = \frac{\sin \left(x\right) + 1}{\cos} \left(x\right)$

Then just simplify to get:
$\frac{\sin \left(x\right) + 1}{\cos} \left(x\right) = \frac{\sin \left(x\right) + 1}{\cos} \left(x\right)$

And there you have it.

See explanation.

#### Explanation:

Proving trigonometric identities means to show that one side is equal to the other.

You can do it in three ways:

a. Start with the LHS and show that it equals to the RHS.
b. Start with the RHS and show that it equals to the LHS.
c. Work with both sides simultaneously until you arrive at the same expression for both.

WARNING: Some teachers like myself love to give exercises where some identities are false. So how can you avoid wasting your time trying to prove such identities true, when in fact they are false and can't be proved?

You try substituting an obscure angle (say, ${17.314}^{\setminus} \circ$) and checking if both sides are equal. If they are not, then don't waste your time because you have found a counterexample . Stop and write: Can't be proved because it is false.

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