Steps to solve:
5 - 5cos(π/2 - x) = 2cos²(π - x)
~Use an identity to simplify cos(π/2 - x)
cos(π/2 - x) → sin(x)
~Use an identity to simplify cos(π - x)
cos(π - x) → -cos(x)
~Put back into an expression
5 - 5sin(x) = 2(-cos(x))²
~Simplify
5 - 5sin(x) = 2cos²(x)
~Subtract 2cos²(x) to both sides
5 - 5sin(x) - 2cos²(x) = 2cos²(x) - 2cos²(x)
~Simplify
5 - 5sin(x) - 2cos²(x) = 0
~Use an identity to simplify
5 - (1 - sin²(x)) * 2 - 5sin(x) = 0
~Simplify
3 + 2sin²(x) - 5sin(x) = 0
~Let sin(x) = u
3 + 2u² - 5u = 0
~Use the quadratic formula to solve for u.
[tex]x=\frac{-(-5)+-\sqrt{(-5)^2-4(2)(3)} }{2(2)}[/tex]
[tex]u = \frac{3}{2}, u=1[/tex]
~Substitute back with sin(x) = u
sin(x) = 3/2 or sin(x) = 1
Steps to solve:
log₂(1 - x) + log₂(-5x - 2) = 2 + log₂3
~Apply log rule
log₂((1 - x(-5x - 2)) = 2^2+(log₂(3)
~Use log definition
(1 - x)(-5x - 2) = 2^2+log₂(3)
~Expand both sides
5x² - 3x - 2 = 12
~Subtract 12 to both sides
5x² - 3x - 2 - 12 = 12 - 12
~Simplify
5x² - 3x - 14 = 0
~Use the quadratic formula to solve
[tex]x = \frac{-(-3)+-\sqrt{(-3)^2-4(5)(-14)} }{2(5)}[/tex]
~Simplify
x = 2 or x = -7/5
Best of Luck!
