Please find the roots for the equation 
 
 
 

Asked by rajuharan | 7th Jun, 2016, 10:14: PM

Expert Answer:

C o n s i d e r space t h e space g i v e n space d e t e r m i n a n t
open vertical bar table row cell sin x end cell cell cos x end cell cell cos x end cell row cell cos x end cell cell sin x end cell cell cos x end cell row cell cos x end cell cell cos x end cell cell sin x end cell end table close vertical bar equals 0
E x p a n d i n g space t h e space d e t e r m i n a n t comma space w e space h a v e comma
sin x open parentheses sin squared x minus cos squared x close parentheses minus cos x open parentheses cos x sin x minus cos squared x close parentheses plus cos x open parentheses cos squared x minus cos x sin x close parentheses equals 0
rightwards double arrow s i n x open parentheses s i n squared x minus c o s squared x close parentheses minus 2 cos x open parentheses c o s x s i n x minus c o s squared x close parentheses equals 0
rightwards double arrow sin cubed x minus sin x cos squared x minus 2 cos squared x sin x plus 2 cos cubed x equals 0
rightwards double arrow s i n cubed x minus 3 c o s squared x s i n x plus 2 c o s cubed x equals 0
D i v i d i n g space t h e space e q u a t i o n space b y space cos cubed x comma space w e space h a v e comma
tan cubed x minus 3 tan x plus 2 equals 0
L e t space t equals tan x
T h e n space t h e space a b o v e space e q u a t i o n space b e c o m e s comma space w e space h a v e comma
t cubed minus 3 t plus 2 equals 0
F a c t o r i sin g space t h e space a b o v e space e q u a t i o n space b e c o m e s comma space w e space h a v e comma
open parentheses t minus 1 close parentheses open parentheses t minus 1 close parentheses open parentheses t plus 2 close parentheses equals 0
rightwards double arrow t equals 1 space a n d space t equals negative 2
rightwards double arrow A s space t equals negative 2 space i s space n o t space a space v a l i d space o n e comma space w e space h a v e comma space t equals 1
rightwards double arrow tan x equals 1
rightwards double arrow x equals tan to the power of negative 1 end exponent open parentheses 1 close parentheses
rightwards double arrow x equals straight pi over 4
H e n c e space t h e r e space i s space o n l y space o n e space r e a l space r o o t space s a t i s f y i n g space t h e space g i v e n space e q u a t i o n.

Answered by Vimala | 8th Jun, 2016, 04:29: PM