234 4 Geometry and Trigonometry
657.Show that the trigonometric equation
sin(cosx)=cos(sinx)
has no solutions.658.Show that if the anglesaandbsatisfy
tan^2 atan^2 b= 1 +tan^2 a+tan^2 b,
then
sinasinb=±sin 45◦.659.Find the range of the functionf:R→R,f(x)=(sinx+ 1 )(cosx+ 1 ).
660.Prove that
sec^2 nx+csc^2 nx≥ 2 n+^1 ,
for all integersn≥0, and for allx∈( 0 ,π 2 ).
661.Compute the integral
∫ √
1 −x
1 +x
dx, x∈(− 1 , 1 ).662.Find all integerskfor which the two-variable functionf (x, y)=cos( 19 x+ 99 y)
can be written as a polynomial in cosx,cosy,cos(x+ky).
663.Leta, b, c, d∈[ 0 ,π]be such that
2 cosa+6 cosb+7 cosc+9 cosd= 0
and
2 sina−6 sinb+7 sinc−9 sind= 0.
Prove that 3 cos(a+d)=7 cos(b+c).664.Letabe a real number. Prove that
5 (sin^3 a+cos^3 a)+3 sinacosa= 0. 04
if and only if
5 (sina+cosa)+2 sinacosa= 0. 04.665.Leta 0 ,a 1 ,...,anbe numbers from the interval( 0 ,π 2 )such that
tan(
a 0 −
π
4)
+tan(
a 1 −
π
4)
+···+tan(
an−
π
4)
≥n− 1.Prove that
tana 0 tana 1 ···tanan≥nn+^1.