SATHEE: Trigonometrical Ratios and Identities 1 Question 21

Trigonometrical Ratios and Identities 1 Question 21

21. For a positive integer $n$ , let

$\begin{aligned} f_{n} (θ) = \tan \frac{θ}{2} (1 + \sec θ) (1 + \sec 2 θ) \\ (1 + \sec 2^{2} θ) \dots (1 + \sec 2^{n} θ), then \end{aligned}$

(1999, 3M)

(a) $f_{2} \frac{π}{16} = 1$

(b) $f_{3} \frac{π}{32} = 1$

(d) $f_{5} \frac{π}{128} = 1$

Match the Column

Match the conditions/expressions in Column I with values in Column II.

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Solution:

NOTE Multiplicative loop is very important approach in IIT Mathematics.

$\begin{aligned} \begin{aligned} \tan \frac{θ}{2} (1 + \sec θ) & = \frac{\sin θ / 2}{\cos θ / 2} \cdot 1 + \frac{1}{\cos θ} \\ = \frac{(\sin θ / 2) 2 \cos^{2} θ / 2}{(\cos θ / 2) \cos θ} \\ = \frac{(2 \sin θ / 2) \cos θ / 2}{\cos θ} = \frac{\sin θ}{\cos θ} = \tan θ \\ ∴ f_{n} (θ) & = (\tan θ / 2) (1 + \sec θ) \\ (1 + & \sec 2 θ) (1 + \sec 2^{2} θ) \dots (1 + \sec 2^{n} θ) \\ = (\tan θ) (1 + \sec 2 θ) (1 + \sec 2^{2} θ) \dots (1 + \sec 2^{n} θ) \\ = \tan 2 θ \cdot (1 + \sec 2^{2} θ) \dots (1 + \sec 2^{n} θ) \\ = \tan (2^{n} θ) \end{aligned} \end{aligned}$

Now, $f_{2} \frac{π}{16} = \tan 2^{2} \cdot \frac{π}{16} = \tan \frac{π}{4} = 1$

Therefore, (a) is the answer.

$f_{3} \frac{π}{32} = \tan 2^{3} \cdot \frac{π}{32} = \tan \frac{π}{4} = 1$

Therefore, (b) is the answer.

$f_{4} \frac{π}{64} = \tan 2^{4} \cdot \frac{π}{64} = \tan \frac{π}{4} = 1$

Therefore, (c) is the answer.

$f_{5} \frac{π}{128} = \tan 2^{5} \cdot \frac{π}{128} = \tan \frac{π}{4} = 1$

Therefore, (d) is the answer.

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अगला

Trigonometrical Ratios and Identities 1 Question 21

21. For a positive integer $n$ , let

Match the Column

Solution:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

Trigonometrical Ratios and Identities 1 Question 21

21. For a positive integer n, let

Match the Column

Solution:

इंजीनियरिंग की तैयारी

चिकित्सा तैयारी

एनसीईआरटी पुस्तकें और समाधान

महत्वपूर्ण संसाधन

अन्य संसाधन

पाठ्यक्रम

परीक्षा मंच

नमूना मॉक टेस्ट

सदस्यता

एनसीईआरटी व्यायाम वीडियो समाधान

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21. For a positive integer $n$ , let