Respuesta :
[tex] (\cos x-\frac{\sqrt{2}}{2})(\sec x-1) [/tex]=0 [/tex]
[tex] =(\cos x-\frac{1}{\sqrt{2}})(\sec x-1) [/tex]=0 [/tex]
[tex] \frac{(\sqrt{2}\cos x-1)}{\sqrt{2}}(\frac{1}{\cos x\ }-1)=0 [/tex]
(Reciprocal Identity)
[tex] (\frac{^{\sqrt{2}\\cos x-1}}{^{\sqrt{2}}})(\frac{1-\cos x}{\cos x})=0 [/tex]
[tex] \frac{^{(\sqrt{2}\cos x-1})}{\sqrt{2}}\frac{(1-\cos x)}{\cos x}=0 [/tex]
[tex] (\sqrt{2}\cos x-1}){(1-\cos x)}=0 [/tex] (ZeroProduct Property)
[tex] \sqrt{2}\cos x-1=0 [/tex]
[tex] \sqrt{2}\cos x=1 [/tex]
[tex] \cos x=\frac{1}{\sqrt{2}} [/tex]
[tex] x=\frac{\Pi }{4} [/tex]
and
[tex] 1-\cos x=0 [/tex]
[tex] \cos x=1 [/tex]
[tex] x=0 [/tex]
x=0 and x=[tex] \frac{\Pi }{4} [/tex] are the solutions.
Answer and explanation :
Given : [tex](\cos x-\frac{\sqrt{2}}{2})(\sec x-1)=0[/tex]
To find :
I. Use the zero product property to set up two equations that will lead to solutions to the original equation.
Solution :
The zero product property state that,
If [tex]x\times y=0[/tex] then x=0 or y=0 (or both x=0 and y=0)
Applying zero product property we get,
[tex](\cos x-\frac{\sqrt{2}}{2})(\sec x-1)=0[/tex]
[tex](\cos x-\frac{\sqrt{2}}{2})=0\text{ or }(\sec x-1)=0[/tex]
The two equations form is
[tex]\cos x-\frac{\sqrt{2}}{2}=0[/tex]....(1)
[tex]\sec x-1=0[/tex] ......(2)
II. Use a reciprocal identity to express the equation involving secant in terms of sine, cosine, or tangent.
Solution :
The reciprocal identity is flipping of a number,
The reciprocal of secant is 1 by cosine
[tex]sec x=\frac{1}{cos x}[/tex]
Substitute in the given equation,
[tex](\cos x-\frac{\sqrt{2}}{2})(\frac{1}{cos x}-1)=0[/tex]
III. Solve each of the two equations in Part I for x, giving all solutions to the equation
Solution :
The two equations form is
[tex]\cos x-\frac{\sqrt{2}}{2}=0[/tex]....(1)
[tex]\sec x-1=0[/tex] ......(2)
Solving equation (1)
[tex]\cos x-\frac{\sqrt{2}}{2}=0[/tex]
[tex]\cos x-{1}\frac{\sqrt{2}}=0[/tex]
[tex]\cos x={1}\frac{\sqrt{2}}[/tex]
[tex]\cos x=\cos \frac{\pi}{4}[/tex]
[tex]x=\frac{\pi}{4}[/tex]
Solving equation (2)
[tex]\sec x-1=0[/tex]
[tex]\sec x=1[/tex]
[tex]\sec x=\sec 0[/tex]
[tex]x=0[/tex]
Therefore, The solutions of the equation is [tex]x=0,\frac{\pi}{4}[/tex]
