Question

Why is sec(-x) equivalent to sec x?

Answer 1

secx is 1/cosx and cosx is an even function, meaninging cos(-x) = cos(x)

Answer 2

-x is in quadrant IV , x is in quadrant I, they are both positive for cosine , so as sec

Answer 3

What of cot and cosec?

Answer 4

In the unit circle, cosx >0 in the quadrand I and IV => cox = cos( -x)

Answer 5

cosec -> quadrant I, II
cot -> quadrant I, III

Answer 6

See the above drawing. In the fourth quadrant, the distance AB is negative(measured in -y direction) while AC is positive(in +y direction)
cos(-x)=OA/OB
cos(x)=OA/OC
Now, OC=OB(From congruency of triangles OAC and OAB)
so cos(-x)=cosx
similarly for secx(just the reciprocal)
Do similarly for cot(-x) and cosec(-x), you will find that they are not equal to cotx and cosecx