1. What does it actually mean to verify a trig identity?
When we verify a trig identity this means that we are proving that this trig identity always true. This identity should always be true no matter what is plugged into it if it is used with its rightful pair. sec and tan go together which means that they can turn into each other back and forth with the edition of the one.Since sec^2x - tan^2x= 1. This can be arranged in all sorts of ways and it will be true since it is and identity meaning that is has been proven that this equation will always work. Cos and Sin go together making cos^2x + sin^2 = 1 and it too can be arranged into different forms and it will always be true because these identities have been verified that it works. Cot and csc go together making 1+ cot^2x= csc^2x. All these identities have been verified that they are true and that will always work if they are being used in the correct form.
2. Tips and tricks i have found helpful.
Some tips and tricks that we can use in order to simplify or solve trig equations are that we can always check for a GCF( greatest common factor), conjugate, breaking down fractions that are monomials, factoring, foiling, and substitution of identities. (Kirch SSS packet). If you are completely stuck you can always change everything in terms of sin and cos. Most of the time if there is a potential way to make and identity or to form one that is the path i decide to take. The reason being that this usually tends to cancel with something else or it can become the one trig function that is left to use. When we see that we have a binomial either in the numerator or the denominator and there is no given identity then we can always multiply by a conjugate. This will help us get squares and lead us to finding an identity. When we choose to factor is is because we have either the same base or because we have multiples of 2 on our exponent. When we foil is because we know that there is a possibilities that we may get squares making that a potential identity that can simplify to something else. The reason why we decide to change everything to sin and cos if we are completely stuck is to see if there is a different rout that we end up checking and work away around it.
3. Verifying a trig function
To verify a trig function it means that we are not allowed to touch the second part of the equal sign. What we are trying to do is make sure that this is valid and will somehow transform into that on the other side. There are multiple ways of being able to verify this but they all have to tie back to the one that is on the other side. The side that we shall always work is usually the more complicated side. When we verify a trig function we state that this is right and that somehow it will turn to what was given to us, giving us a true statement. The first thing that we can try to look at is see what we can do to change the complicated side into the trig function(s) that are listed on the other side of the equal sign. We do this by seeing if there is a GCF that will factor out, if there is and identity. If these are not there then we go to our other options being foiling which i would only use if there are to trig functions that could become and identity if they were being squared. The next option would be finding the least common multiple which would be used if we have fractions that we would like to combine together. Our next option would be using the conjugate this would be used if we have a binomial either on the numerator of on the denominator and we would foil this out. Our last option would be change everything to sin and cos. This is mostly used if you cannot find any other way to verify the trig function in other to make it the same as what is on the other side of the parenthesis. There are times where you feel like that is all you can do in terms of simplifying the answer and it is just not the answer that is on the other side. See if you can break it down if it is a fraction, or see if there is any other way that it can be addressed using ratio identities or reciprocal identities. If you don't get it on your first attempt keep trying because there are multiply ways of verifying a trig function, you will eventually get there. :)
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