The general solutions of the equation ^2 + 2 =1 are

Step-by-step solution

Transforming the equation ^2 + 2 = 1 begins with the identity ^2 = ^2 - 1 , yielding: ^2 + 2 = 2 Expressing secants as reciprocals of cosines, 1 ^2 + 1 2 = 2 , and using 2 = 2 ^2 - 1 to write ^2 = 1 + 2 2 , substitution gives: 2 1 + 2 + 1 2 = 2 Combining fractions with a common denominator results in 3 2 + 1 2 (1 + 2 ) = 2 . Cross-multiplying and simplifying leads to the quadratic equation: 2 ^2 2 - 2 - 1 = 0 Factoring as (2 2 + 1)( 2 - 1) = 0 gives solutions 2 = - 1 2 or 2 = 1 . For 2 = 1 , the general solution is