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Both identities are proven as follows:
These proofs show that both equations hold true for all values of
where the expressions are defined.
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Let’s prove the first identity:
To begin, we cross-multiply both sides:
On the right side, we can use the difference of squares:
We know from the Pythagorean identity that:
Thus:
This means the left side equals the right side, confirming the identity:
Now, let’s prove the second identity:
Start with the left side. Factor the numerator:
Recall
:
So we rewrite the numerator:
The denominator can be simplified as well:
Thus, our left side becomes:
Next, rewriting the right side:
Through our simplification process, characterized above, we’ve shown that both sides are indeed equal:
Thus, both identities hold true!