Solve, algebraically or with justification (no graphing calculator solutions allowed), the following equation: tan7x-sin6x=cos4x-cot7x
To solve the equation tan7x - sin6x = cos4x - cot7x algebraically, we'll use some trigonometric identities and simplification techniques. Let's go step by step:
Step 1: Simplify the equation using trigonometric identities.
Recall the following trigonometric identities:
- tanθ = sinθ/cosθ
- cotθ = cosθ/sinθ
Using these identities, we can rewrite the equation as:
sin7x/cos7x - sin6x = cos4x - cos7x/sin7x
Step 2: Combine similar terms.
We can rearrange the terms to put them on the same side:
sin7x/cos7x - cos7x/sin7x - sin6x + cos4x = 0
Step 3: Find a common denominator.
To combine the fractions, we need a common denominator. The common denominator for cos7x and sin7x is cos7x * sin7x. Multiplying the respective terms by this common denominator, we get:
(sin7x)^2 - (cos7x)^2 - sin6x * cos7x * sin7x + cos4x * cos7x * sin7x = 0
Step 4: Apply trigonometric identities and simplify further.
We can substitute the following trigonometric identities:
- (sin7x)^2 - (cos7x)^2 = sin^2x(1 - cos^2x) = sin^2x * sin^2(6x)
- sin6x * cos7x * sin7x = sin6x * sin7x * cos7x = sin6x * sin7x * (1 - sin^2x)
- cos4x * cos7x * sin7x = cos4x * sin7x * cos7x = sin7x * cos4x * (1 - sin^2x)
Substituting these identities into the equation, we get:
sin^2x * sin^2(6x) - sin6x * sin7x * (1 - sin^2x) + sin7x * cos4x * (1 - sin^2x) = 0
Step 5: Simplify the equation further.
Expanding the equation, we have:
sin^2x * sin^2(6x) - sin6x * sin7x + sin6x * sin^2x * sin7x + sin^2x * sin7x * cos4x - sin^2x * sin7x * cos4x = 0
Step 6: Factor out common terms.
Factoring out sin6x * sin7x and sin^2x, we get:
sin6x * sin7x (sin^2x - 1) + sin^2x * sin7x (cos4x - 1) = 0
Step 7: Find solutions.
For the equation to be equal to zero, either sin6x * sin7x = 0 or (sin^2x - 1) = 0 or (cos4x - 1) = 0.
When sin6x * sin7x = 0, it means that either sin6x = 0 or sin7x = 0.
For sin6x = 0, x can take any value such that 6x = nπ where n is an integer.
For sin7x = 0, x can take any value such that 7x = nπ where n is an integer.
When (sin^2x - 1) = 0, it means that sin^2x = 1.
For sin^2x = 1, x can take any value such that x = nπ/2 where n is an integer.
When (cos4x - 1) = 0, it means that cos4x = 1.
For cos4x = 1, x can take any value such that 4x = 2nπ where n is an integer.
These are the potential solutions to the equation. To obtain specific values of x, you can substitute values for n and solve for x in each case.
Note: Please review and check for extraneous solutions as we have used several simplifications and substitutions throughout the process.