tanx-cotx/tanx+cotx=1-2cos squardx
To solve the equation tan(x) - cot(x) / tan(x) + cot(x) = 1 - 2cos^2(x), we'll simplify each side step-by-step:
Step 1: Rewrite tan(x) and cot(x) using sine and cosine:
Using the identities tan(x) = sin(x) / cos(x) and cot(x) = cos(x) / sin(x), we rewrite the equation as follows:
(sin(x) / cos(x)) - (cos(x) / sin(x)) / (sin(x) / cos(x)) + (cos(x) / sin(x)) = 1 - 2cos^2(x)
Step 2: Simplify the denominators:
Let's simplify the denominator of the left side by finding a common denominator for (sin(x) / cos(x)) and (cos(x) / sin(x)):
(sin(x) / cos(x)) - (cos(x) / sin(x)) / [(sin(x) * sin(x)) / (cos(x) * sin(x))] + [(cos(x) * cos(x)) / (cos(x) * sin(x))]
This becomes:
(sin(x) / cos(x)) - (cos(x) / sin(x)) / [sin^2(x) / cos(x)sin(x)] + [cos^2(x) / cos(x)sin(x)]
Step 3: Simplify further:
Using the reciprocal property of division, the expression becomes:
(sin(x) / cos(x)) - (cos(x) / sin(x)) * [cos(x)sin(x) / sin^2(x)] + [cos^2(x) / cos(x)sin(x)]
(sin(x) - cos^2(x)) / cos(x)sin(x) + cos^2(x) / cos(x)sin(x)
Step 4: Combine the fractions:
To combine the fractions, we need a common denominator, which is cos(x)sin(x). The expression becomes:
[(sin(x) - cos^2(x)) + cos^2(x)] / cos(x)sin(x)
(sin(x) - cos^2(x) + cos^2(x)) / cos(x)sin(x)
Step 5: Simplify the numerator:
The numerator can be simplified:
sin(x) - cos^2(x) + cos^2(x)
sin(x)
Step 6: Final result:
The equation simplifies to:
sin(x) / cos(x)sin(x) = 1 - 2cos^2(x)
1 / cos(x) = 1 - 2cos^2(x)
Multiplying both sides by cos(x):
1 = cos(x) - 2cos^3(x)
Rearranging the terms:
2cos^3(x) - cos(x) + 1 = 0
Therefore, the simplified equation is 2cos^3(x) - cos(x) + 1 = 0.
To solve the given trigonometric equation:
tan(x) - cot(x) / tan(x) + cot(x) = 1 - 2cos²(x)
We can use trigonometric identities to simplify and manipulate the equation.
Let's start by using the definitions of tangent and cotangent:
tan(x) = sin(x) / cos(x)
cot(x) = cos(x) / sin(x)
Substitute these values into the equation:
(sin(x) / cos(x)) - (cos(x) / sin(x)) / (sin(x) / cos(x)) + (cos(x) / sin(x)) = 1 - 2cos²(x)
Next, simplify the expression:
(sin(x)sin(x) - cos(x)cos(x)) / (cos(x)sin(x) + sin(x)cos(x)) = 1 - 2cos²(x)
Using the Pythagorean identity sin²(x) + cos²(x) = 1, simplify further:
(1 - cos²(x) - cos²(x)) / (cos(x)sin(x) + sin(x)cos(x)) = 1 - 2cos²(x)
Simplify the numerator:
(1 - 2cos²(x)) / (cos(x)sin(x) + sin(x)cos(x)) = 1 - 2cos²(x)
Now, let's simplify the denominator:
cos(x)sin(x) + sin(x)cos(x) = 2cos(x)sin(x)
Substitute back into the equation:
(1 - 2cos²(x)) / (2cos(x)sin(x)) = 1 - 2cos²(x)
Multiply both sides by 2cos(x)sin(x) to eliminate the denominator:
1 - 2cos²(x) = (1 - 2cos²(x))(2cos(x)sin(x))
Expand the right side:
1 - 2cos²(x) = 2cos(x)sin(x) - 4cos³(x)sin(x)
Rearrange the terms:
1 = 2cos(x)sin(x) - 4cos³(x)sin(x) + 2cos²(x)
Combine like terms:
1 = 2cos(x)sin(x) - 4cos³(x)sin(x) + 2cos²(x)
Now, let's factor out a sin(x) common from the right side:
1 = sin(x)(2cos(x) - 4cos³(x) + 2cos²(x))
Now, consider the following identity: sin(x) = 1 - cos²(x)
Replace sin(x) in the equation:
1 = (1 - cos²(x))(2cos(x) - 4cos³(x) + 2cos²(x))
Multiply through by -1 to simplify:
-1 = cos²(x)(4cos³(x) - 2cos²(x) - 2cos(x) + 1)
Simplify further:
cos²(x)(4cos³(x) - 2cos²(x) - 2cos(x) + 2) = 1
Divide both sides by cos²(x):
4cos³(x) - 2cos²(x) - 2cos(x) + 2 = 1 / cos²(x)
Rearrange the terms:
4cos³(x) - 2cos²(x) - 2cos(x) + 2 - 1 / cos²(x) = 0
Now, we have a cubic equation in terms of cosine (cos(x)). To solve this equation, we can use numerical methods or graphing techniques.
Unfortunately, there is no simple algebraic solution for this equation.