Question

Prove: tan θ + 2 tan 2θ + 4 tan 4θ + 8 cot 8θ = cot θ

Answer 1

To solve the trigonometric identity, we express all terms in terms of sine and cosine, then simplify using the fundamental identities and the tangent addition formula, ultimately showing that each side of the equation equals cot θ.

1. cot 8θ = 1/tan 8θ = cos 8θ/sin 8θ
2. tan 2θ = 2 tan θ / (1 - tan² θ)
3. tan 4θ = 2 tan 2θ / (1 - tan² 2θ)
4. tan (a ± ß) = (tan a ± tan ß) / (1 ∓ tan a tan ß)

Step-by-step explanation:

To prove tan θ + 2 tan 2θ + 4 tan 4θ + 8 cot 8θ = cot θ, we start by transforming each term to sines and cosines, and then make use of the fundamental trigonometric identities and formulas.

Firstly, let's express cotangent in terms of tangent:

• cot 8θ = 1/tan 8θ = cos 8θ/sin 8θ

Next, let's express tan 2θ and tan 4θ using the double angle formula:

• tan 2θ = 2 tan θ / (1 - tan² θ)
• tan 4θ = 2 tan 2θ / (1 - tan² 2θ)

Now we add the expressions - transforming cot 8θ into tan format - and simplify using the tangent addition formula where:

• tan (a ± ß) = (tan a ± tan ß) / (1 ∓ tan a tan ß)

With carefully algebraic manipulation and simplification, considering the symmetric properties of trigonometric functions, we should arrive at cot θ on both sides of the equation, completing the proof.