SOLUTION: verifying trigonometric identities. (2-sin^2x/cosx)=secx+cosx

Question 592917: verifying trigonometric identities.
(2-sin^2x/cosx)=secx+cosx
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
These identities can be hard because there is no "recipe" that one can memorize and then apply to each one.

When I looked at your problem here is what I saw:

The right side is expressed in terms of sec(x) and cos(x)
Since sec(x) is the reciprocal of cos(x) it might be helpful to find a way to express the left side in terms of cos(x)
The left side is expressed in terms of sin^2(x) and cos(x). It is very easy to replace the sin^2(x) using sin^2(x) + cos^2(x) = 1 or sin^2(x) = 1 - cos^2(x)
The right side is two terms while the left side is just one. So at some point we will need to find a way to split the one term into two.

So let's put these ideas into action:

Replacing sin^2(x) with 1-cos^2(x) we get:

Note the use of parentheses! This is an extremely good habit (in any expression, not just Trig expressions) whenever making substitutions involving different numbers of terms. (Here we are replacing 1 term with two.)

Simplifying the numerator carefully we get:

(I hope it is clear why there is a "+" in front of the cos^2(x) and how it would be very easy to get this wrong without the parentheses!)

Now we will split the fraction into to terms (by "un-adding" them):

The first fraction is sec(x) and we can cancel a factor of cos(x) in the second fraction leaving us with:
sec(x) + cos(x) = sec(x) + cos(x)
And we're done!
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