Plane trigonometry, by S.L. Loney.

EXAMPLES.

477

MISCELLANEOUS EXAMPLES. LXIX.
1. Prove that the equation tan x= kx has an infinite number of roots.
2. If A, B and C be the angles of a triangle, prove that
1- 8 cos A cos B cos C
is always positive.
3. If a and 3 be the imaginary cube roots of unity prove that
ax Px - - J3x /3x
aea + e(P = e 2 sin 2- cos ]
2-3 cos x
4. If x be less than a radian prove that x 2/\/ cos x very
5 + cos x
x5
nearly, the error in the left-hand member being nearly 480 radians.
5. If cos (0 + i)=sec (a+ if3), where a, /, 0, and 5 are all real, prove
that
tanh2 q cosh2 - sin2 a and tanh2 3 cosh2 = sin2 0.
6. If x = 2 cos a cosh / and y = 2 sin a sinh 3,
prove that
4x
sec (a + i3) + see (a - ip) = 2 + 2
4iy
and see (a + i/) - sec (a - i3) = + 2
7. Prove that
sin' 90 cos nO + n sinn-1 0 cos (n - 1) 0 sin (0 - ~)
+ n ( sinn-2 0 cos (n - 2) 0 sin2 (0 - ) +...... sin ( - )
1.2
= sin' 0 cos n7.
8. Prove that the roots of the equation
n (n - 1) 2
x sin nO - nxn-l sin (nO ) + 0) 2 X) -2 sin (nO + 25)
-...... to (n+ 1) terms=0,
are given by x=sin(0 + - ) cose 0 - k ),
where n is an integer and k has any integral value from 0 to n- 1.