Jim tries to create a logarithmic-trigo problem. He takes the following steps to create the problem:

- Let \(p\) and \(q\) be distinct prime numbers.
- Then, let \((\log_{10}q)^5=\sin^2p+\cos^2p\), where \(p\) is in degrees.
- Find all pairs \((p,q)\) satisfying the above conditions.
- According to him, there are infinite such pairs, as \(p\) can take infinite values to make the RHS of the equation equal to 1.

Has he created the problem with the solution correctly?

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