SOLUTION: You are supposed to show the steps it takes to get the answer: (sec˛x - 6 tan x + 7)/ (sec˛x - 5) = (tan x - 4)/ (tan x + 2)

Algebra ->  Trigonometry-basics -> SOLUTION: You are supposed to show the steps it takes to get the answer: (sec˛x - 6 tan x + 7)/ (sec˛x - 5) = (tan x - 4)/ (tan x + 2)      Log On


   

Question 993812: You are supposed to show the steps it takes to get the answer:
(sec˛x - 6 tan x + 7)/ (sec˛x - 5) = (tan x - 4)/ (tan x + 2)
Found 2 solutions by Alan3354, KMST:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
The answer to what question?

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!

I assume you want to find the value(s) of x that is/are solutions to the equation above.
Solutions must make the expressions above exist.
The functions sec%28x%29=1%2Fcos%28x%29 and tan%28x%29=sin%28x%29%2Fcos%28x%29 must be defined,
and the denominators must not be zero:
sec%5E2%28x%29+-+5%3C%3E0%7D%7D%3C--%3E%7B%7B%7Bsec%5E2%28x%29%3C%3E5<-->1%2Fcos%5E2%28x%29%3C%3E5<-->cos%5E2%28x%29%3C%3E1%2F5 and
tan%28x%29+%2B+2%3C%3E0<-->tan%28x%29%3C%3E-2
So far, we know that the solution is such that
system%28cos%28x%29%3C%3E0%2Ccos%5E2%28x%29%3C%3E1%2F5%2Ctan%28x%29%3C%3E-2%29 .
For any other value of x , we try to simplify and solve.
There may be better ways, but this is what I came up with:



1+%2B+%2812-6tan%28x%29%29%2F%28sec%5E2%28x%29-5%29+=+1+-+6%2F%28tan%28x%29%2B2%29
%2812-6tan%28x%29%29%2F%28sec%5E2%28x%29-5%29+=+-+6%2F%28tan%28x%29%2B2%29
6%282-tan%28x%29%29%2F%28sec%5E2%28x%29-5%29+=+-+6%2F%28tan%28x%29%2B2%29
%282-tan%28x%29%29%2F%28sec%5E2%28x%29-5%29+=+-+1%2F%28tan%28x%29%2B2%29
%28tan%28x%29-2%29%2F%28sec%5E2%28x%29-5%29+=+1%2F%28tan%28x%29%2B2%29<-->%28tan%28x%29%2B2%29%28tan%28x%29-2%29=sec%5E2%28x%29-5<-->tan%5E2%28x%29-4=sec%5E2%28x%29-5<--><-->sin%5E2%28x%29-4cos%5E2%28x%29=1-5cos%5E2%28x%29<-->1-cos%5E2%28x%29-4cos%5E2%28x%29=1-5cos%5E2%28x%29<-->1-5cos%5E2%28x%29=1-5cos%5E2%28x%29 .
The last equation is an identity.
So, the answer is any x such that system%28cos%28x%29%3C%3E0%2Ccos%5E2%28x%29%3C%3E1%2F5%2Ctan%28x%29%3C%3E-2%29 .