Expert: Sherry Wallin - 10/14/2009

Question
QUESTION: 1+tan^2
_______
1-tan^2

ANSWER: What is it you want done with this Fasha?

Usually instructions come with problems. Does it say simplify or does it say show that it is equal to some other thing? I will assume they want it simplified so here is how you would do it:

First, all trig functions have to have an argument, i.e., an angle and you have 1+tan^2 which is incorrect, you must say 1+tan^2x where x is the angle. The 2 by the way means squared, so 1+tan^2x is one plus tan squared on the angle x.

1-tan^2x is just sec^2x (an identity) so you have 1/sec^2x + tan^2x/sec^2x = cos^2x + {sin^2x/cos^2x)/(1/cos^2x) since sec^2x = 1/cos^2x. You now have cos^2x + sin^2x = 1 also an identity. So the expression [1+ tan^2x]/[1-tan^2x] is just 1.

Math Prof

---------- FOLLOW-UP ----------

QUESTION: verify the following identities
a) cot^2x - tan^2x = cosec^2x - sec^2x
b)(tan A + tan B )/ (cot A + cot B )= tan A tan B
c)which one is correct?
 1+tan^2x = sec^2x or 1-tan^2X = sec^2x

Answer
Fasha~
    To verify you need to simplify both sides until you have the same expression on both sides and then write it as if you were going from one to the other. I will do one for you. Before I forget to do part c simplify both sides of each and see if/when they have the same expression and that will tell you which is correct.

a) cot^2x - tan^2x = cosec^2x - sec^2x Let's start with cot^2x - tan^2x and see where it takes us:

cot^2x - tan^2x = cos^2x/sin^2x - sin^2x/ cos^2x. Get a common denominator:

[cos^2x(cos^2x)-sin^2x(sin^2x)]/sin^2xcos^2x = [cos^4x-sin^4x]/sin^2xcos^2x

= (cos^2x + sin^2x)(cos^2x-sin^2x)/sin^2xcos^2x = 1(cos^2x-sin^2x)/sin^2xcos^2x


start with csc^2x - sec^2x and see where it takes us:


csc^2x - sec^2x = 1/sin^2x -1/cos^2x. Get a common denominator:

[cos^2x - sin^2x]/sin^2xcos^2x this is the same thing we got above so you would write the whole

problem like this:


cot^2x - tan^2x

= cos^2x/sin^2x - sin^2x/ cos^2x

= [cos^2x(cos^2x)-sin^2x(sin^2x)]/sin^2xcos^2x

= [cos^4x-sin^4x]/sin^2xcos^2x

= 1/sin^2x -1/cos^2x

= csc^2x - sec^2x


Math Prof