1.) To verify trig functions it means to prove that the equation on the right is true by showing that the left side equals to the right. In other words, taking an equation and simplifying it until it is exactly the same as the right side. Verifying trig functions can be challenging because there is no set or "correct" method to verify these trig function and all it comes down is to a pick and guess situation and if it takes you to a more difficult path, then you can reverse and try another method.
2.) The number one tip I could give is listen to Mrs. Kirch when she says that you need to study in order to remember all the trig ratios, identities, and Pythagorean identities. The most helpful trick I can give is if you are stuck and you can get things to be converted into sin and cos then you should take that route. Another, tip I can give is to substitute the variables into something that is different from all the sin, pluses, minuses,etc. because a + next to a tan can be confusing.
3.)When verifying trig functions the first thing that I think is can I take out a GCF. If I can then you might want to do that first because perhaps things start cancelling out right there and then. The next step I take is to see if I can substitute an identity. It can be one of the ratio identities, reciprocal identities, or Pythagorean identities. Then, I look if the denominator is a binomial and if it is I can multiply by the conjugate to the numerator and denominator. Another thing you can perform is to combine fractions with binomial denominator or separate fraction, but only when the denominator is a monomial. In addition, you can factor and if all fails you can resort to changing everything to sin and cos.