To solve the equation [tex]\(\frac{2}{\sin A + \cos A + 1} = 2 \sec A\)[/tex], let's go through the steps in detail:
1. Rewrite Secant Function: Recall that [tex]\(\sec A\)[/tex] is the reciprocal of [tex]\(\cos A\)[/tex]:
[tex]\[
\sec A = \frac{1}{\cos A}
\][/tex]
2. Substitute [tex]\(\sec A\)[/tex] into the Equation: Substitute [tex]\(\sec A = \frac{1}{\cos A}\)[/tex] into the given equation:
[tex]\[
\frac{2}{\sin A + \cos A + 1} = 2 \cdot \frac{1}{\cos A}
\][/tex]
3. Simplify the Right-Hand Side: Simplify the right side of the equation:
[tex]\[
\frac{2}{\sin A + \cos A + 1} = \frac{2}{\cos A}
\][/tex]
4. Eliminate the Denominator: Since both sides of the equation have a denominator, we can cross-multiply to eliminate the fractions:
[tex]\[
2 \cos A = 2 (\sin A + \cos A + 1)
\][/tex]
5. Simplify the Equation: Divide both sides by 2 to simplify:
[tex]\[
\cos A = \sin A + \cos A + 1
\][/tex]
6. Isolate [tex]\(\sin A\)[/tex]: Subtract [tex]\(\cos A\)[/tex] from both sides to isolate [tex]\(\sin A\)[/tex]:
[tex]\[
0 = \sin A + 1
\][/tex]
[tex]\[
\sin A = -1
\][/tex]
7. Determine the Angle: We need to find the angle [tex]\(A\)[/tex] where [tex]\(\sin A = -1\)[/tex]. The sine function equals [tex]\(-1\)[/tex] at the angle:
[tex]\[
A = 270^\circ \text{ or } \frac{3\pi}{2} \text{ radians}
\][/tex]
Hence, the solution to the equation [tex]\(\frac{2}{\sin A + \cos A + 1} = 2 \sec A\)[/tex] is:
[tex]\[
A = 270^\circ
\][/tex]
