Solve-for-real-numbers-sin-x-cos-x-sec-x-csc-x-2-2-

Question Number 157265 by MathSh last updated on 21/Oct/21

Solve for real numbers :

\sin (x) + \cos (x) + \sec (x) \cdot \csc (x) = 2 + \sqrt{2}

Answered by TheSupreme last updated on 21/Oct/21

s i n (x) + c o s (x) - \frac{1}{s i n (x) c o s (x)} = A

\pm s \sqrt{1 - s^{2}} - (\pm \frac{1}{s \sqrt{1 - s^{2}}}) = A

s^{2} \sqrt{1 - s^{2}} + s - s^{3} = \pm A s \sqrt{1 - s^{2}}

(s^{2} - (\pm A s)) \sqrt{1 - s^{2}} = s^{3} - s

s \neq 0

(s - (\pm A)) \sqrt{1 - s^{2}} = s^{2} - 1

(s - (\pm A)) = - \sqrt{1 - s^{2}}

s^{2} + A^{2} - (\pm 2 s A) = 1 - s^{2}

2 s^{2} - (\pm 2 s A) + A^{2} - 1 = 0

s_{1, 2} = \frac{2 A \pm \sqrt{4 A^{2} + 8 A^{2}}}{4} = A \frac{1 \pm \sqrt{3}}{2} (a)

s_{1, 2} = A \frac{- 1 \pm \sqrt{3}}{2}

\dots

s ⩽ 1

s_{1} = (2 + \sqrt{2}) (\frac{1 - \sqrt{3}}{2}) (a)

c o s = \sqrt{1 - s^{2}}

s_{2} = (2 + \sqrt{2}) (\frac{- 1 + \sqrt{3}}{2}) (b)

c o s = - \sqrt{1 - s^{2}}