Starting with one of our ‘sum and difference’ identities:

If we take

then, using the identity we started with:

We can do it all again with cos:

Now, recall that

so

That’s one identity, now if we go back to the start and manipulate it all a bit differently:

Finally, tan is pretty straight forward if you use the sum&difference identity, but we haven’t derived that yet, so here we go:

Using the sin & cos sum/difference identities:

Phew! Now, if x and y are equal, then just call them both x:

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*Related*

The identities come naturally from Euler’s formula. See the website: https://betterexplained.com/articles/easy-trig-identities-with-eulers-formula/.