We have now seen what the trigonometric functions are and where they appear in real life. Let's now dive more into more algebraic-type problems of trigonometric functions.

An **equation** is a statement that two mathematical expressions are equal. For example, the following are equations
\begin{align}
x + 2 &= 5\\
(x + 1)^2 &= x^2 + 2x + 1\\
\sin^2 x + \cos^2 x &= 1
\end{align}
An **identity** is an equation that is true for all the values of the variables in the equation. For example, in the above equation $\sin^2 x + \cos^2 x = 1$ is true for any values of $x$; we saw this was true in Section 5.2.

This section will heavily depend on identities that were introduced in Section 5.2. Make sure to keep these in your back pocket and use these often!

The first type of problem we will encounter is to simplify a complicated looking expression.

Simplfy the expression $\cos t + \tan t \sin t$.

Simplify the expression $\dfrac{\sin\theta}{\cos \theta} + \dfrac{\cos \theta}{1 + \sin \theta}$.

Strategies:

- Try to convert trigonometric functions into sines and cosines (problem 1).
- Try to find a common denominator (problem 1 and 2).

We have two subproblems.

If someone walks up to you and says **"All horses are brown"**, to disprove this statement, you only need to find one example of a horse which is not brown, such as a zebra.

To disprove an identity, we use the same idea. For example, given **$\sin x + \cos x = 1$**, to disprove this statement, we only need to find one $x$ which makes this equation false (recall identity means true for all variable values).

Show the equation $\sin x + \cos x = 1$ is not an identity.

Here are a few guidelines to prove an equation is actually an identity. They are not necessarily taken in order.

**Start with one side.**Given one side, try to transform it into the other side. Try to pick one side that "looks more complicated", such as involving many sums and products.**Use known identities**from Section 5.2.- Find a common denominator.
**Convert to sines and cosines.**If you are stuck, try to convert all functions to sines and cosines. Perhaps then you can apply the previous tactics.- Introduce something extra by multiplying by 1. Transform that 1 into whatever you want introduced.
- Choose both the LHS and RHS. Simplify both, and you should get the same result.

To prove an identity, **only perform reversible operations.** Non-reversible operations include

- Squaring both sides.
- Taking square roots of both sides.

Given \[\sin x = - \sin x\] try squaring both sides. What happens?

In general, you **cannot** start with a false initial statement, get to a true conclusion, then conclude that initial statement is true. A similar concept also appears in daily conversations and is called Argument from False Premises.

Verify algebraically that \[\cos\theta (\sec \theta - \cos \theta) = \sin^2 \theta\]
Also verify graphically.

Verify this identity algebraically \[2\tan x \sec x = \frac{1}{1 - \sin x} - \frac{1}{1 + \sin x}\]

Verify this identity algebraically \[\frac{\cos \theta}{1 - \sin \theta} = \sec \theta + \tan \theta\]

Verify \[\frac{1 + \cos \theta}{\cos \theta} = \frac{\tan^2 \theta}{\sec \theta - 1}\]

Assume $0 \leq \theta \leq \pi/2$. Substitute $\sin \theta$ for $x$ in the expression $\sqrt{1 - x^2}$ and simplify.