Respuesta :
Recall the triple angle identity for cosine:
cos(3x) = cos³(x) - 3 sin²(x) cos(x)
… = cos³(x) - 3 (1 - cos²(x)) cos(x)
… = 4 cos³(x) - 3 cos(x)
and the definition of secant,
sec(x) = 1/cos(x)
So we have
sec(x) cos(3x) = 0
(4 cos³(x) - 3 cos(x))/cos(x) = 0
cos(x) (4 cos²(x) - 3)/cos(x) = 0
If cos(x) ≠ 0 (this happens at the endpoints of the interval [-π/2, π/2]), we can simplify this to
4 cos²(x) - 3 = 0
cos²(x) = 3/4
cos(x) = ±√3/2
But since -π/2 < x < π/2, we know cos(x) > 0, so we ignore the negative case:
cos(x) = √3/2
==> x = π/6 and x = -π/6
The solution of the given trigonometric equation by using trigonometric identities is [tex]\frac{\pi }{6} \ and \frac{-\pi }{6}[/tex].
What are trigonometry identities?
Trigonometric Identities are the equalities that involve trigonometry functions and holds true for all the values of variables given in the equation.
Some trigonometric identities are
[tex]cos(3x) = 4cos^{3}(x) -3cos(x)[/tex]
According to the given question.
We have an equation
[tex]sec(x)cos(3x) = 0[/tex]
Since, the above equation can be written as by using trigonometric identities
[tex]sec(x)cos(3x) = 0\\\implies \frac{1}{cos(x)} (4cos^{3} x-3cos(x))=0[/tex]
Solve the above equation for x.
[tex]\implies 4cos^{2} x -3= 0[/tex]
[tex]\implies 4cos^{2}x = 3\\ \implies cos^{2} x = \frac{3}{4} \\\implies cos x = \sqrt{\frac{3}{4} } \\\implies cos x = \pm\frac{\sqrt{3} }{2}[/tex]
In the given domain [tex][\frac{-\pi }{2}, \frac{\pi }{2} ][/tex] we know that cosx > 0. Therefore, we take only positive part
[tex]\implies cosx = \frac{\sqrt{3} }{2} \\\implies x = cos^{-1} \frac{\sqrt{3} }{2} \\\implies x = \frac{\pi }{6}, and \frac{-\pi }{6}[/tex]
Hence, the solution of the given trigonometric equation by using trigonometric identities is [tex]\frac{\pi }{6} \ and \frac{-\pi }{6}[/tex].
Find out more information about trigonometric identities here:
https://brainly.com/question/12537661
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