Expert: Sherry Wallin - 4/6/2009

Question
tan+(cos/1+sin)=sec
plz solve this for me and send me the answer back asap

Answer
Sean~
    I will help you but you need to practice finding these relationships because it is important to develop your skills in manipulating definitions and identities for further work in mathematics. I think you have written your relationship incorrectly though. I think you mean tan x + cos x/(1+sin x) = sec x. Hopefully you notice that you left off the arguments for each of the aforementioned trig functions and it is NEVER ok to write a trig function without an argument [in this case I just called the argument x]. Also the use of parentheses,if you are meaning to divide the cos x by the whole quantity of sin x + 1 it must be written the way I did. Now onto showing the identity: Notice I did not say solve because this is a 'show' problem. There are many ways to approach this.One way is to work both sides of the equal sign and try to arrive at the same expression, another is to start with one side and try to arrive at the other side. Let's try this last method first and if we find it too difficult we can always use the other method.

tan x + cos x/(1+sinx) = sin x/cos x + cos x/(1+ sin x)
=  sin x/cos x + [cos x(1-sin x)]/[(1+sin x)(1-sin x)]

{never underestimate the power of multiplying a fraction by the denominators conjugate, this is what I did when I multiplied top and bottom by the conjugate of 1-sin x which is 1+ sin x}

= sin x/cos x + [cos x(1-sin x)]/[1-sin x]^2
[1-sin x]^2 is [cos x]^2

= sin x/cos x + [cos x(1-sin x)]/[cos x]^2
notice on the right I can cancel the cos x on top with one of the factors of [cos x]^2 since [cos x]^2 = cos x * cos x

= sin x/cos x + (1-sin x)]/cos x  and now you have an expression that has cos x in both denominators

= [sin x + 1 - sinx]/cos x
= 1/cos x
= sec x

Notice I didn't do any unnecessary math,i.e., I didn't multiply out
[cos x(1-sin x)] until I saw if I needed to or not. I also didn't put the expression over a common denominator, I waited to see if I needed to. You can see I didn't need to do either of those things and it still worked out just fine.

Practice and practice a lot this is how you get good at it...

Math Prof