Question

Find one value of θ or x that satisfies each of the following.tan θ = cot(45° + 2θ)

Accepted Answer

Recall the definition of cotangent in terms of tangent: $$\cot \alpha = \frac{1}{	an \alpha}. So $$the equation $$	an 	heta = \cot(45^\circ + 2	heta)$$ can be rewritten as $$	an 	heta = \frac{1}{	an(45^\circ + 2	heta)}. $$Multiply both sides of the equation by $$	an(45^\circ + 2	heta) to $$get rid of the fraction: $$	an 	heta \cdot 	an(45^\circ + 2	heta) = 1. $$Use the tangent addition formula to express $$	an(45^\circ + 2	heta)$$: $$	an(a + b) = \frac{	an a + 	an b}{1 - 	an a 	an b}. $$Here, $$a = 45^\circ$$ and $$b = 2	heta$$, so $$	an(45^\circ + 2	heta) = \frac{	an 45^\circ + 	an 2	heta}{1 - 	an 45^\circ 	an 2	heta}. $$Substitute $$	an 45^\circ = 1$$ into the expression to simplify: $$	an(45^\circ + 2	heta) = \frac{1 + 	an 2	heta}{1 - 	an 2	heta}. $$Replace $$	an(45^\circ + 2	heta) in $$the equation $$	an 	heta \cdot 	an(45^\circ + 2	heta) = 1$$ with the simplified expression and solve for $$	heta$$: $$	an 	heta \cdot \frac{1 + 	an 2	heta}{1 - 	an 2	heta} = 1. $$From here, you can proceed to isolate $$	heta.$$

Find one value of θ or x that satisfies each of the following.
tan θ = cot(45° + 2θ)

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Key Concepts

Relationship Between Tangent and Cotangent

Angle Sum Identities

Solving Trigonometric Equations