The number of roots of the equation cos7x - sin4x = 1 that lie in the interval [ 0 , - Maths - Inverse Trigonometric Functions

Question

The number of roots of the equation cos7 x -
sin4 x = 1 that lie in the interval [ 0 , 2pi ]
is:
(a) 2 (b) 3 (c) 4 (d) 8

please give a detailed answer of this question

Ghanshyam Dhakar · Accepted Answer

Dear
student,
cos7x-sin4x=1put sin2x=1-cos2xcos7x-1-cos2x2=1cos7x-1-cos4x+2cos2x=1cos7x-cos4x+2cos2x-2=0let cosx=tt7-t4+2t2-2=0   
1by hit and trial methodput t=11-1+2-2=00=0so t=1 is root of  eq1t-1t6+t5+t4+2t+2=0t-1t4t2+t+1+2t+1=0 t4t2+t+1>0 always and we know -1 ≤t≤1 then 2t+1≥0so t4t2+t+1+2t+1 >0 , no solution then  t=1 is root of  eq1cosx  =1x=0, 2π answer a 
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The number of roots of the equation cos7 x - sin4 x = 1 that lie in the interval [ 0 , 2pi ] is: (a) 2 (b) 3 (c) 4 (d) 8 please give a detailed answer of this question.

The number of roots of the equation cos⁷ x - sin⁴ x = 1 that lie in the interval [ 0 , 2pi ] is:
(a) 2 (b) 3 (c) 4 (d) 8

please give a detailed answer of this question.