To solve the equation cos^2(x) + sin(x) = √2sin(x+π/4), we can start by using the Pythagorean identity sin^2(x) + cos^2(x) = 1.
Rewrite the given equation in terms of sin and cos using the Pythagorean identity:
cos^2(x) + sin(x) = √2sin(x+π/4)1 - sin^2(x) + sin(x) = √2sin(x+π/4)
Now, we can rearrange terms and simplify:
sin(x) - sin^2(x) = √2sin(x+π/4)sin(x)(1 - sin(x)) = √2sin(x+π/4)
Now, let's use the sum to product identity for the right side:
sin(x)(1 - sin(x)) = √2(sin(x)cos(π/4) + cos(x)sin(π/4))sin(x)(1 - sin(x)) = √2(sin(x)(1/√2) + cos(x)(1/√2))sin(x)(1 - sin(x)) = sin(x) + cos(x)
Since sin(x) is a common factor, we can divide by sin(x):
1 - sin(x) = 1 + cos(x)
Now, solving for cos(x) in terms of sin(x):
cos(x) = -sin(x)
Hence, cos(x) = -sin(x) is the solution to the given equation cos^2(x) + sin(x) = √2sin(x+π/4).
To solve the equation cos^2(x) + sin(x) = √2sin(x+π/4), we can start by using the Pythagorean identity sin^2(x) + cos^2(x) = 1.
Rewrite the given equation in terms of sin and cos using the Pythagorean identity:
cos^2(x) + sin(x) = √2sin(x+π/4)
1 - sin^2(x) + sin(x) = √2sin(x+π/4)
Now, we can rearrange terms and simplify:
sin(x) - sin^2(x) = √2sin(x+π/4)
sin(x)(1 - sin(x)) = √2sin(x+π/4)
Now, let's use the sum to product identity for the right side:
sin(x)(1 - sin(x)) = √2(sin(x)cos(π/4) + cos(x)sin(π/4))
sin(x)(1 - sin(x)) = √2(sin(x)(1/√2) + cos(x)(1/√2))
sin(x)(1 - sin(x)) = sin(x) + cos(x)
Since sin(x) is a common factor, we can divide by sin(x):
1 - sin(x) = 1 + cos(x)
Now, solving for cos(x) in terms of sin(x):
cos(x) = -sin(x)
Hence, cos(x) = -sin(x) is the solution to the given equation cos^2(x) + sin(x) = √2sin(x+π/4).