To find the numerical values of [tex]x[/tex], we are going to solve the trigonometric equation [tex]tan^2(x)- \frac{sin^2(x)}{sin^2(x)}= 5[/tex] for [tex]x[/tex].
Step 1. Simplify, [tex]\frac{sin^2(x)}{sin^2(x)}=1[/tex]:
[tex]tan^2(x)- \frac{sin^2(x)}{sin^2(x)}= 5[/tex]
[tex]tan^2(x)- 1= 5[/tex]
Step 2. Add 1 to both sides of the equation:
[tex]tan^2(x)- 1+1= 5+1[/tex]
[tex]tan^2(x)=6[/tex]
Step 3. Take square root to both sides of the equation:
[tex] \sqrt{tan^2(x)}=(+/-) \sqrt{6} [/tex]
[tex]tan(x)= \sqrt{6} ,- \sqrt{6} [/tex]
Step 4. Solve for [tex]x[/tex]:
[tex]x=arctan( \sqrt{6} )+k \pi ,x=arctan(- \sqrt{6} )+k \pi [/tex]
[tex]x=1.1832+k \pi ,x=-1.1832+k \pi [/tex]
We can conclude that the numerical values of [tex]x[/tex] that satisfy the equation are [tex]x=1.1832+k \pi [/tex] and [tex]x=-1.1832+k \pi[/tex]
