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If it is ggiven that sin 2 x + 3cosx-2 = 0, then find the value of cos 3 x + sec 3 x

If it is ggiven that sin2x + 3cosx-2 = 0, then find the value of cos3x + sec3x

Grade:10

2 Answers

Arun
25763 Points
4 years ago
Dear Student
sin^2x = 1 – cos^2 x
now
1 – cos^2 x + 3 cos x – 2=0
cos^2 x- 3cos x +1 = 0
let cos x = t
then it will form a quadratic equation
t^2 – 3t + 1 = 0
t = (3 + \sqrt5)/2  and t = (3 –  \sqrt5)/2
1st value is cancelled
now 
cos x= (3 –  \sqrt5)/2
now you can easily find out the answer.
 
regards
Arun (askIITians forum expert)
Rishabh Sankla
33 Points
2 years ago
Dear Student,
Hope this will help you.
 
sin^2 x + 3 cos x = 2
 
\Rightarrow (1-cos^2 x) + 3 cos x = 2
 
\Rightarrow cos^2 x - 3 cos x = -1                [After shifting terms]
 
\Rightarrow cos x(cos x - 3) = -1
 
\Rightarrow cos x - 3 = -sec x
 
\Rightarrow cos x + sec x = 3
 
\Rightarrow (cos x + sec x)^3 = (3)^3
 
\Rightarrow cos^3 x + sec^3 x + 3 cos x sec x(cos x + sec x) = 27
 
\Rightarrow cos^3 x + sec^3 x + 3\times3 = 27
 
\Rightarrow cos^3 x + sec^3 x = 18
 
This is the simpler method to find answer to this question.
 
Regards
Rishabh

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