What is sec 180 degrees?

-1 See the answer is quite simple We know that in 1st quadrant all trig. functions are +ve In 2nd quadrant only sin/cosec are +ve In 3rd tan/cot are +ve and In 4th quadrant cos/sec are +ve.

Now have a look at this table.

From here we are able to know that the value of cos(0)=1.

Now look at the circle.0 degrees means it makes 0 degrees with the X-axis(we always measure angles with reference to x-axis).So 180 degrees with the x-axis will result the angle being in 2nd quadrant. But in 2nd quadrant only Sin/cosec are +ve therefore the cos/sec of amy angle here would be negative.Since 0 and 180 degrees both lie on the x-axis the value of cos(180) would be same as that of cos(0) but with a "-" sign. Now for sec,we know that reciprocal of cos is sec Therefore sec(180)=-1

Hope this helped!

P.S:This is not how exactly you get the values of trig. functions. Actually we draw the radius of the circle making the required angle with the x-axis and then use trignomerty basics to find out values.