# How do I Use Mathematica to Prove a Trigonometric Identity?

Grandson and I are working on integrating trigonometric functions and he asked if Mathematica could be used to "prove" a trigonometic identity. We looked over various functions and it looked like "Equal" would be our best bet.

However, while "Equal", represented by two == signs, works when faced with simple identities such as:

 Tan[alpha] == 1 / Cot[alpha]

True


it seems to "fail" when given more complex identities such as

 Sec[alpha]^4 - Sec[alpha]^2 == Tan[alpha]^4 + Tan[alpha]^2.


In those cases, it simply repeats the input as the output with no other other indication as to why, etc.

We've tried this with several other known identities with the same result.

Is there a better function for this? Or, are we asking too much of Mathematica, etc.

• Try Simplify[Sec[alpha]^4 - Sec[alpha]^2 == Tan[alpha]^4 + Tan[alpha]^2] Sometimes that might not be enough and you might need to tell Simplify that alpha is Real or positive or an integer or ... – Bill Jul 9 '18 at 4:43
• I would not say "prove". I would say "check". As in "How do I Use Mathematica to Check a Trigonometric Identity?" – Lotus Jul 9 '18 at 10:28
• == is not exactly for testing equality. It will only returns True if the LHS and RHS are literally identical. Otherwise, it simply stands to represent an equality which may be true for all, some or no values of the variables. What happened here is that 1 / Cot[x] immediately evaluated to Tan[x], making the two sides not only mathematically but also literally identical. – Szabolcs Jul 9 '18 at 11:40

Sec[alpha]^4 - Sec[alpha]^2 == Tan[alpha]^4 + Tan[alpha]^2 // Simplify

Reduce[ForAll[x, Sin[x] == 1/Cos[x]], Reals]