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.