How do you simplify the expression #sec(arctan ((2x)/5))#?

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Answer 1 · 2016-08-22T12:25:18

#sqrt(1+4/15)x^2#

Let #a = arc tan ((2/5)x) in Q1 or Q4#, according as # x > or < 0#.

Then, #tan a = 2/5x, and sec > 0#, in both Q1 and Q4.

Now, the given expression is

#sec a = sqrt(1+tan^2a)=sqrt(1+4/15)x^2#.

If a is assumed to be the general value, it might #in Q3#, for negative

x. In this case, sec a becomes negative and the answer becomes

#+-sqrt(1+4/25x^2),#..