Correct Answer – Option 1 : cot θ 

Concept:

Trigonometric Identities:

• \(\rm \tan\theta=\dfrac{\sin\theta}{\cos\theta}\).
• \(\rm \cot\theta=\dfrac{\cos\theta}{\sin\theta}\).
• sin² θ + cos² θ = 1.
• sin 2θ = 2 sin θ cos θ.
• cos 2θ = cos² θ – sin² θ.

Calculation:

Let us observe that:

\(\rm \cot 2\theta=\dfrac{\cos 2\theta}{\sin2\theta}=\dfrac{\cos^2\theta-\sin^2\theta}{2\sin\theta\cos\theta}=\dfrac{1}{2}(\cot\theta-\tan\theta)\)

⇒ cot θ – tan θ = 2 cot 2θ             … (1)

⇒ tan θ = cot θ – 2 cot 2θ             … (2)

Now, tan θ + 2 tan 2θ + 4 tan 4θ + 8 cot 8θ

= (cot θ – 2 cot 2θ) + 2 tan 2θ + 4 tan 4θ + 8 cot 8θ             … Using equation (2)

= cot θ – 2(cot 2θ – tan 2θ) + 4 tan 4θ + 8 cot 8θ

= cot θ – 2(2 cot 4θ) + 4 tan 4θ + 8 cot 8θ             … Using equation (1)

= cot θ – 4(cot 4θ – tan 4θ) + 8 cot 8θ

= cot θ – 4(2 cot 8θ) + 8 cot 8θ             … Using equation (1)

= cot θ – 8 cot 8θ + 8 cot 8θ

= cot θ.

• sin (A ± B) = sin A cos B ± sin B cos A.
• cos (A ± B) = cos A cos B ∓ sin A sin B.