SOLUTION: Find all solutions of the equation tan²x + 2sec²x = 7, where -<font face="symbol">p</font>/2 < x < <font face="symbol">p</font>/2.

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Question 438002: Find all solutions of the equation tan²x + 2sec²x = 7, where -p/2 < x < p/2.
Answer by Edwin McCravy(20065)   (Show Source): You can put this solution on YOUR website!


      tan²x + 2sec²x = 7

Replace sec²x by 1 + tan²x

tan²x + 2(1 + tan²x) = 7

Distribute:

  tan²x + 2 + 2tan²x = 7

Combine like terms:

          2 + 3tan²x = 7

Subtract 2 from both sides:

              3tan²x = 5

Divide both sides by 3

               tan²x = 

Use the principle of square roots

               tan x = 

Rationalize the denominator:

               tan x = 

               tan x =   
                        
Since calculators give principle values for the inverse
trig functions, and since -p/2 < x < p/2 is the range for 
tan-1.

One solution is tan-1()
                  
which is x = 0.911738291 

The other solution is tan-1()
                  
which is x = -0.911738291
Edwin
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