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Question

Total number of solutions of equation sin x tan 4x = cos x belonging to (0,π) are:


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Solution

In this case also we have to simplify the given expression. For that we will write tan 4x = sin4xcos4x

We do this to apply the formula for sin(A±B) or cos(A±B) after bringing the terms to one side.
sin4x sin4xcos4x = cosx
Multiplying by cos 4x on both the sides.

sinxsin4x=cosxcos4x(cos4x0)

cos4xcosxsinxsin4x=0

cos(4x+x)=0

cos5x=0

5x = (2n + 1)π2
x = (2n + 1)π10
Putting n = 1, 2, 3 we get,
x = π10 ,3π10 ,5π10 ,7π10 ,9π10
the next value 11π10 — is out of our required range (0,π).

we get 5 solutions.


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