1-sin2x/cosx=tan(π/4-x)
as written, it is not true. I think you meant
(1-sin2x)/cos2x
= (1 - 2sinx cosx)/(cos^2x - sin^2x)
= (cos^2x - 2sinx cosx + sin^2x)/(cos^2x - sin^2x)
= (cosx-sinx)^2 / ((cosx-sinx)(cosx+sinx))
= (cosx-sinx)/(cosx+sinx)
= (1-tanx)/(1+tanx)
= tan(π/4 - x)
To simplify the given equation, we'll use trigonometric identities and simplify each side separately.
Starting with the left-hand side:
1 - sin(2x)/cos(x)
We can rewrite sin(2x) using the double-angle identity:
sin(2x) = 2sin(x)cos(x)
Now, we can substitute that back into the original equation:
1 - (2sin(x)cos(x))/cos(x)
Cancel out the cos(x) terms:
1 - 2sin(x)
Now let's simplify the right-hand side:
tan(π/4 - x)
Applying the tangent identity, we have:
tan(π/4 - x) = (sin(π/4 - x))/(cos(π/4 - x))
Using the sum-to-product identities, we can write:
sin(π/4 - x) = sin(π/4)cos(x) - cos(π/4)sin(x)
Since sin(π/4) = cos(π/4) = 1/√2, we can substitute those values:
(1/√2)(cos(x)) - (1/√2)(sin(x))
Combining the terms, we get:
(1/√2)(cos(x) - sin(x))
Now, equating the left-hand side and the right-hand side, we can rewrite the equation as:
1 - 2sin(x) = (1/√2)(cos(x) - sin(x))
To solve for x, we'll isolate the terms with sin(x) and cos(x).
First, distribute the (1/√2) to the terms in the right-hand side:
1 - 2sin(x) = (1/√2)cos(x) - (1/√2)sin(x)
Next, move the terms with sin(x) to the left-hand side, and the terms with cos(x) to the right-hand side:
2sin(x) - (1/√2)sin(x) = (1/√2)cos(x) - 1
Combining like terms:
(2 - 1/√2)sin(x) = (1/√2)cos(x) - 1
Simplifying the coefficient of sin(x):
((2√2 - 1)/√2)sin(x) = (1/√2)cos(x) - 1
Dividing through by √2:
(2√2 - 1)sin(x) = cos(x) - √2
Now, we can solve for x by isolating sin(x) and cos(x) on opposite sides:
(2√2 - 1)sin(x) = cos(x) - √2
Multiply both sides by cos(x):
(2√2 - 1)sin(x)cos(x) = cos²(x) - √2cos(x)
Using the identity sin²(x) + cos²(x) = 1, we can rewrite the equation as:
(2√2 - 1)sin(x)cos(x) = 1 - √2cos(x)
Move all terms to one side:
(2√2 - 1)sin(x)cos(x) + √2cos(x) - 1 = 0
Factor out cos(x):
cos(x)[(2√2 - 1)sin(x) + √2 - 1] = 0
Now we have two possibilities:
1) cos(x) = 0
2) (2√2 - 1)sin(x) + √2 - 1 = 0
To solve these equations, you'll need to solve for x in each case by using appropriate trigonometric methods or a calculator.