Given that:
sec x+tan x=22/7.

Determine the value of:
csc x + cot x

sec x + tan x = 22/7
(1 + sinx)/cosx = 22/7
7(1 + sinx) = 22(cosx)
49(1 + 2sinx + sin^2(x)) = 484 cos^2(x)
add 484 sin^2(x) to both sides
49 + 98sin(x) + 533sin^2(x) = 484
533sin^2(x) + 98sin(x) - 435 = 0

sin(x) = [-98 ± sqrt(9604 + 927420)]/1066
note that 937024 = 968^2

sin(x) = [-98 + 968]/1066 = 870/1066 = 0.8161
     or  [-98 - 968]/1066 = 1066/1066 = 1
The arcsin(0.8161) = 0.954690764747344 radians
                   = 54.6997515601398 degrees
Plug into original equation: 3.14285714285714 = 22/7  check

The arcsin(1) = 1.5707963267949 radians
              = 90 degrees
Plug into original equation: divide by zero; reject

x = 0.954690764747344 radians
csc x + cot x  = 1.93333333333333 = 29/15

In summary:
sec + tan = 22/7
csc + cot = (22+7)/(22-7)  
Conclusion:  there must have been a simpler solution

Posted by Larry on 2023-10-27 12:31:53