Michael H. answered • 07/23/19

All levels of undergraduate mathematics and statistics

Remember that tan(x – y) = (tan x – tan y)/(1 + tan x tan y).

Therefore tan(x – π/4) = (tan x – tan(π/4))/(1 + tan x tan π/4).

And tan(π/4) = 1, so tan(x – π/4) = (tan x – 1)/(1 + tan x).

Since you have sin(x) = 7/25, you have cos(x) = ±√(1 – (7/25)^{2}) = ±24/25, and since x is in the first quadrant, you conclude that cos(x) = 24/25.

Then tan(x) = sin(x) / cos(x) = 7/24.

So, finally, tan(x – π/4) = (tan x – 1)/(1 + tan x) = (7/24 – 1)/(1 + 7/24)

= (7 – 24)/(24 + 7) = –17/31.

(If you're not following the above, one could wonder if it's because your posted question did not specify which aspects of the problem you had difficulty with. Normally you should do that.)