Question
Question: The number of solutions for the equation \(\sin x.\tan 4x=\cos x\) in \(\left( 0,\pi \right)\) is\[\...
The number of solutions for the equation sinx.tan4x=cosx in (0,π) isA.4
B.7C.8
D. None of these $$$$
Solution
We replace tan4x=cos4xsin4x in the given equation and use the cosine sum of two angles formula cos(A+B)=cosAcosB−sinAsinB to make equation like cosθ=0 whose solutions we determine in the form of θ=(2n+1)2π. We check how many the solutions lie in the gives interval (0,π).$$$$
Complete step by step answer:
We know that the solutions of the equation cosx=0 are the odd integral multiples of 2π that is x=(2n+1)2π where n is an integer. We also know the cosine sum of two angle formula where the cosine of sum of two angles say A and B is given by
cos(A+B)=cosAcosB−sinAsinB
The given equation is
sinx.tan4x=cosx
We know that for any angle θ , tanθ=cosθsinθ. We use it and replace tan4x in the above equation to get,
sinx.cos4xsin4x=cosx
We multiply cos4x in both side of the equation and get