Find the exact value of the expression Please dont use a cal

Find the exact value of the expression. Please don\'t use a calculator

Find the exact value of the expression. sec(- pi/3) -cot(- 5pi/4)

Solution

3 ) sec(-pi/4) -cot(-5pi/4)

Property : cos(-x) = cosx and tan(-x) = -tanx ; 1/cosx = secx ; 1/tanx = cotx

So, sec(-pi/4) -cot(-5pi/4) = 1/cospi/4 +1/tan5pi/4

Now tan5pi/4 = tan(pi +pi/4) = tanpi/4

tan(pi/4) =1 and cospi/4 = 1/sqrt2

So, sec(-pi/4) -cot(-5pi/4) = 1/cospi/4 +1/tan5pi/4

=1/cospi/4 +1/tanpi/4

=sqrt2 +1