Solving Trigonometric Equations: From sin^3 x cot x cos^3 x tan x cos 2x to sin x cos x 1 - sin x cos x
Objective: In this article, we will solve the trigonometric equation sin^3 x cot x cos^3 x tan x cos 2x using step-by-step algebraic manipulation and trigonometric identities. By working through this problem, we will explore methods that involve simplification, identity usage, and solution verification.
Problem Statement
We start with the equation:
sin^3 x cot x cos^3 x tan x cos 2x
Step-by-Step Solution
1. Simplification and Identity Application
First, let's expand and simplify the left-hand side (LHS) of the equation:
sin^3 x 1/cot x cos^3 x 1/tan x sin^3 x (cos x/sin x) cos^3 x (sin x/cos x)
Multiplying and simplifying further:
sin^3 x (cos^4 x/sin x) (sin^2 x/cos x)
Combining the terms:
sin^2 x cos^2 x sin x cos x sin x cos x
2. Right-Hand Side (RHS) Simplification
Now, let's simplify the right-hand side (RHS) using the double-angle identity:
cos 2x cos^2 x - sin^2 x
Expanding the RHS:
cos 2x (cos x - sin x)(cos x sin x)
This simplifies to:
sin x cos x cos x - sin x (cos x sin x)
Which simplifies to:
sin x cos x cos x - sin x
3. Further Simplification and Identity Usage
Next, we cancel the terms and square both sides to find the solutions:
cos x - sin x 1
Squaring both sides:
(cos x - sin x)^2 1^2
This simplifies to:
cos^2 x - 2 sin x cos x sin^2 x 1
Using the Pythagorean identity cos^2 x sin^2 x 1, we get:
1 - 2 sin x cos x 1
Solving for sin x and cos x:
sin x cos x 0
Thus, either:
sin x 0 cos x 04. Specific Solutions
From the solutions above, we find:
When sin x 0 and cos x 1 or cos x -1, then x 0, π, 2π, etc. When cos x 0, then x π/2, 3π/2, etc.Therefore, the specific solutions are:
x 0, π, 2π, π/2, 3π/2, etc.
Conclusion
In summary, we have solved the trigonometric equation sin^3 x cot x cos^3 x tan x cos 2x. By applying trigonometric identities and simplifying the equation, we identified that sin x 0 or cos x 0. The specific values of x that satisfy the equation are x 0, π, 2π, π/2, 3π/2, etc.