Explanation
To find the general solution, we will simplify the given trigonometric equation step-by-step.
Given Equation:
cotθ+tanθ=2
Step 1: Convert to Sine and Cosine
We know the basic trigonometric identities:
tanθ=cosθsinθandcotθ=sinθcosθ
Substituting these into the equation:
sinθcosθ+cosθsinθ=2
Step 2: Simplify the Fractions
Take the Least Common Multiple (LCM) on the left-hand side:
sinθcosθcos2θ+sin2θ=2
Using the fundamental Pythagorean identity sin2θ+cos2θ=1:
sinθcosθ1=2
Step 3: Apply the Double-Angle Identity
Cross-multiply to rearrange the equation:
1=2sinθcosθ
We recognize the right-hand side as the double-angle formula for sine (sin2θ=2sinθcosθ):
sin2θ=1
Step 4: Find the General Solution
The principal value for which sin2θ=1 is at 2θ=2π.
For the equation sinα=1, the general solution is:
α=2nπ+2π,n∈Z
Substituting 2θ for α:
2θ=2nπ+2π
Divide the entire equation by 2 to solve for θ:
θ=nπ+4π,n∈Z
Correct Answer
The general solution of the equation is Option A:
θ=nπ+4π,n∈Z
