Checking identities in code

What an identity says

An identity is an equation that is true for every angle where both sides make sense. The best known is

• $\sin^2 a + \cos^2 a = 1$.

It comes from the unit circle. The point $(\cos a, \sin a)$ is on the circle $x^2 + y^2 = 1$.

A program cannot try every angle, but it can try thousands in a blink. The result is useful, but know what it means.

• One failure proves the identity false. You have found a counterexample.
• Many passes do not prove it true. They make it very likely. The proof is in the chapter Trigonometric Functions.

Checking with arrays

Here is a check of $\sin^2 a + \cos^2 a = 1$ on $10\,000$ random angles:

• np.random.default_rng(5) makes a random number generator. The number $5$ is the seed: with the same seed, you get the same angles in every run.
• rng.uniform(0, 2 * np.pi, 10_000) makes $10\,000$ random numbers between $0$ and $2\pi$.
• np.allclose(left, right) is True when every pair of numbers agrees within a tiny tolerance.
• np.max(np.abs(left - right)) is the worst miss. For a true identity it is about $10^{-16}$ to $10^{-15}$. That is rounding error, not a mistake.

The identities to check

These are the identities of the Class 11 chapter. Write each side as a numpy expression, and compare.

• $\sin^2 a + \cos^2 a = 1$.
• $1 + \tan^2 a = \sec^2 a$, where $\sec a = \dfrac{1}{\cos a}$. Avoid angles where $\cos a = 0$.
• $\sin(a + b) = \sin a\cos b + \cos a\sin b$.
• $\cos(a + b) = \cos a\cos b - \sin a\sin b$.
• $\sin 2a = 2\sin a\cos a$ and $\cos 2a = \cos^2 a - \sin^2 a$.

Catching a false identity

A tempting rule is $\sin(a + b) = \sin a + \sin b$. It is false, and a program shows it at once.

To find a counterexample in code, mark the pairs that disagree and pick the first:

~ turns True into False and back. bad.sum() counts the failures, since True counts as $1$. np.argmax of a boolean array is the position of the first True.

The false rule is true for a few special angles, for example when $b = 0$. So one lucky angle proves nothing. Always test many random angles.

Many pairs at once

To test every pair from two lists of angles, make one a column and the other a row. numpy then combines each row with each column:

The shape is $12$ by $12$: one test for each of the $144$ pairs. This is called broadcasting.

Dividing by almost zero

For $1 + \tan^2 a = \sec^2 a$, both sides become huge when $\cos a$ is nearly zero. Then rounding errors are huge too, and np.allclose can fail for a true identity. Test only angles where np.abs(np.cos(a)) > 1e-3.

Practise with new numbers

Try it in Python

The program tests three true identities on $10\,000$ random pairs of angles. It prints the worst miss of each, then tests the false rule and prints a counterexample.

Common mistakes

• Comparing with ==. Rounding errors make a true identity fail. Use np.allclose.
• Testing one angle. A false identity can be true at a few angles. Test thousands.
• Testing only whole degrees. Angles such as $0^\circ$ and $90^\circ$ are special and hide errors. Use random angles.
• Taking a pass as a proof. A program gives strong evidence. A proof needs algebra.
• Ignoring the asymptotes. Skip the angles where the cosine is nearly zero when you test with the tangent or the secant.

Remember these

• An identity holds for every angle. One counterexample proves it false.
• np.allclose(left, right) tests whole arrays. np.max(np.abs(left - right)) is the worst miss, about $10^{-16}$ for a true identity.
• rng = np.random.default_rng(seed) and rng.uniform(0, 2 * np.pi, n) make reproducible random angles.
• np.argmax(bad) of a boolean array is the first True. A column and a row, g[:, None] and g[None, :], test every pair.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A check finds np.allclose(left, right) is False for one angle in a thousand. Is the identity false?

2. Why use random angles and not only the whole degrees $0, 90, 180, 270$?

3. What does np.argmax(bad) return when bad is an array of True and False?

4. Which of $\sin 2a = 2\sin a$ and $\sin 2a = 2\sin a\cos a$ would a program accept?

5. How many pairs does the $12$ by $12$ table of angles test?