Question

║⊕QUESTION⊕║
Why do children dread mathematics? Because of the wrong approach. Because it is looked at as a subject.
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CLASS 11
PRINCIPLE OF MATHEMATICAL INDUCTION
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Using the Principle of Mathematical Induction, prove that
$cos\alpha cos2\alpha cos4\alpha ....cos(2^{n-1}\alpha ) = \frac{sin 2^{n}\alpha }{2^{n}sin\alpha }$ for all n belongs to N

Answer 1

Question

Using Principle of Mathematical Induction,prove that

$cos\alpha cos2\alpha cos4\alpha ....cos(2^{n-1}\alpha ) = \frac{sin 2^{n}\alpha }{2^{n}sin\alpha }$ for all n belongs to N

Solution

Let us assume that P(n) is the given statement and is true for all natural numbers

When n = 1,

LHS :

$\longrightarrow \: \sf{ \cos( \alpha ) }$

RHS :

$\sf{ \dfrac{ \sin(2 \alpha ) }{2 \sin( \alpha ) } } \\ \\ \longrightarrow \: \sf{ \dfrac{2 \sin( \alpha ) \cos( \alpha ) }{2 \sin( \alpha ) } } \\ \\ \longrightarrow \: \sf{ \cos( \alpha ) }$

Thus,P(1) is true

Now,

Let us suppose P(K) is true for all natural numbers,where K is a natural number.

Thus,

$\sf{cos\alpha.cos2\alpha .cos4\alpha ....cos(2^{k-1}\alpha ) = \dfrac{sin {2} {}^{k} \alpha }{2^{k}sin\alpha } - - - - - - (1)}$

To Prove :

P(K + 1) is true

$\sf{cos\alpha. cos2\alpha. cos4\alpha ....cos(2^{k}\alpha ) = \dfrac{sin 2^{k + 1}\alpha }{2^{k + 1}sin\alpha }}$

LHS

$\sf{cos\alpha. cos2\alpha. cos4\alpha ....cos(2^{k}\alpha ) } \\ \\ \longrightarrow \: \sf{cos\alpha. cos2\alpha. cos4\alpha ... \cos( {2}^{k - 1 } \alpha ) .cos(2^{k}\alpha )}$

From Equation (1),we write :

$\sf{\dfrac{ \sin( {2}^{k} ) \alpha }{ {2}^{k} \sin( \alpha ) } \times \cos( {2}^{k} ) \alpha} \\ \\ \longrightarrow \: \sf{\dfrac{sin 2^{k + 1}\alpha }{2^{k + 1}sin\alpha }}$

Thus,LHS = RHS

Hence,P(K + 1) is true whenever P(K) is true

By Principle Of Mathematical Induction,

P(n) is true for all natural numbers

Answer 2

Step-by-step explanation:

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