(cos x - (sqrt 2)/2)(sec x -1)=0 i. use  the  zero  product  property  to  set  up  two  equations  that  will  lead  to  solutions  to  the  original equation. ii. use  a  reciprocal  identity  to  express  the  equation  involving  secant  in  terms  of  sine,  cosine,  or tangent.  iii. solve  each  of  the  two  equations  in  part  i  for  x,  giving  all  solutions  to  the  equation.

Answer and explanation :

Given :

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 then x=0 or y=0 (or both x=0 and y=0)

Applying zero product property we get,

The two equations form is

....(1)

......(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

Substitute in the given equation,

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

Solution :

The two equations form is

....(1)

......(2)

Solving equation (1)

Solving equation (2)

Therefore, The solutions of the equation is

since both sides equal, there are infinitely many olutions.