Answer: [tex]x=n\pi , \frac{3\pi }{4} +n\pi[/tex]
Step-by-step explanation:
Since we have a product here, either term can be zero. We can individually set each term equal to zero:
[tex]tan(x)=0[/tex]
[tex]tan(x)+1=0[/tex]
Beginning with the first term. It is saying "tangent of what gives zero". Well, tangent of zero gives zero. The value pi also gives zero:
[tex]tan(0)=0[/tex]
[tex]tan(\pi )=0[/tex]
We can generalize this by saying 'x' is equal to pi multiplied by an integer 'n', where 'n' is 0, 1, 2, 3...
[tex]x=n\pi[/tex]
Moving on to the second term, lets subtract that 1 to the other side to get:
[tex]tan(x)=-1[/tex]
Tangent of what gives -1? That would be 3pi/4. Tangent is the ratio of sine and cosine. So, you can essentially think of tangent as the slope of the line on a unit circle. Refer to the figure below. Another location along the unit circle where we get -1 is at 7pi/4, which is just 3pi/4 shifted by 180 degrees. Therefore, we can generalize this solution by adding 'npi' to 3pi/4 to get:
[tex]x=3\pi /4+n\pi[/tex]
where 'n' is 0, 1, 2, 3...
