Question

(cos x - (sqrt 2)/2)(sec x -1)=0

Answer 1

$(\cos x-\frac{\sqrt{2}}{2})(\sec x-1)$=0 [/tex]

$=(\cos x-\frac{1}{\sqrt{2}})(\sec x-1)$=0 [/tex]

$\frac{(\sqrt{2}\cos x-1)}{\sqrt{2}}(\frac{1}{\cos x\ }-1)=0$

(Reciprocal Identity)

$(\frac{^{\sqrt{2}\\cos x-1}}{^{\sqrt{2}}})(\frac{1-\cos x}{\cos x})=0$

$\frac{^{(\sqrt{2}\cos x-1})}{\sqrt{2}}\frac{(1-\cos x)}{\cos x}=0$

$(\sqrt{2}\cos x-1}){(1-\cos x)}=0$ (ZeroProduct Property)

$\sqrt{2}\cos x-1=0$

$\sqrt{2}\cos x=1$

$\cos x=\frac{1}{\sqrt{2}}$

$x=\frac{\Pi }{4}$

and

$1-\cos x=0$

$\cos x=1$

$x=0$

x=0 and x=$\frac{\Pi }{4}$ are the solutions.

Answer 2

Answer and explanation :

Given : $(\cos x-\frac{\sqrt{2}}{2})(\sec x-1)=0$

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 $x\times y=0$ then x=0 or y=0 (or both x=0 and y=0)

Applying zero product property we get,

$(\cos x-\frac{\sqrt{2}}{2})(\sec x-1)=0$

$(\cos x-\frac{\sqrt{2}}{2})=0\text{ or }(\sec x-1)=0$

The two equations form is

$\cos x-\frac{\sqrt{2}}{2}=0$....(1)

$\sec x-1=0$ ......(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

$sec x=\frac{1}{cos x}$

Substitute in the given equation,

$(\cos x-\frac{\sqrt{2}}{2})(\frac{1}{cos x}-1)=0$

III. Solve each of the two equations in Part I for x, giving all solutions to the equation

Solution :

The two equations form is

$\cos x-\frac{\sqrt{2}}{2}=0$....(1)

$\sec x-1=0$ ......(2)

Solving equation (1)

$\cos x-\frac{\sqrt{2}}{2}=0$

$\cos x-{1}\frac{\sqrt{2}}=0$

$\cos x={1}\frac{\sqrt{2}}$

$\cos x=\cos \frac{\pi}{4}$

$x=\frac{\pi}{4}$

Solving equation (2)

$\sec x-1=0$

$\sec x=1$

$\sec x=\sec 0$

$x=0$

Therefore, The solutions of the equation is $x=0,\frac{\pi}{4}$