How would I solve the following equation for x?

Question

2 sin^2(x) + 3 tanx secx = 2

I've tried the problems from different approaches, but couldn't come up with a solution. Could you please provide your thought process. It would be greatly appreciated. Thanks!

Answer 1

One way is to use the following identities to transform all terms to sin(x):
sec(x)=1/cos(x)
(1-cos²(x))=sin²(x)

and the resulting quartic equation in sin(x) can be solved. Reject roots whose absolute value exceed 1.

Double check with a plot of the function.

Answer 2

For the solution of the quartic, you should get 0.44 and 1.59 as the approximate real solutions. The latter is rejected because arcsin of 1.59 is undefined.
Thus, solve for x=asin(0.44).

Following link shows the graph:

http://img8.imageshack.us/img8/3576/1285951135.png