60 BOOK OF MATHEMATICAL PROBLEMS. 



2G9. The real roots of the equation tan 8 x tan - = 1 satisfy 

 the equation cos 2x = 2 - J5. 



3 /3 

 270. Given cos 3x = - -jn > prove that the three values of 



f 3 . TT /3 . TT /3 . STT 



cos x are 



aj tan x + a 1 . 



2 < 1 . If the equation tan s = - have real roots, 



2 tan x + a + 1 



a f >l. 



272. Find the limits of the value of * an ( a; + a ) f or possible 



tan (x - a) 



values of x. 



273. If ft y be different values of x given by the equation 



B-y 

 cos ~r-^ m sin (/? + y) = 0. 



2t 



274. The real values of x which satisfy by the equation 

 sin ( | cos x J = cos ( ^ sin x J are 2mr or 2mr ~ , 



n being an integer. 



275. If x, y be real, and if 



sin* x sin 8 y + sin 9 (a; + y) = (sin <e + sin y) 2 , 

 , or y, must be a multiple of IT. 



276. If o, ft y be three angles, unequal and less than 27r, 

 which satisfy the equation 



a b 



- +- + c = 0, 

 cos x sin x 



then will sin (/? + y) + sin (y + a) + sin (a + /3)= 0. 



