`O A ,O Ba n dO C ,w i t hO` as the origin, are three mutually perpendicular lines whose direction cosines are `l_rm_ra n dn_r(r=1,2a n d3)dot` If the projection of `O Aa n dO B` on the plane `z=0` make angles `varphi_1a n dvarphi_2,` respectively, with the x-axis, prove that `tan(varphi_1-varphi_2)=+-n_3//n_1n_2dot`

OA, OB and OC, with O as the origin, are three mutually perpendicular lines whose lines whose direction cosines are l_(r),m_(r)andn_(r)(r=1,2 and 3) . If the projections of OA and OB on the plane z=0 make angles phi_(1)andphi_(2) , respectively, with the x-axis, prove that tan(phi_(1)-phi_(2))=+-n_(3)//n_(1)n_(2) .

If three mutually perpendicular lines have direction cosines (l_1,m_1,n_1),(l_2,m_2,n_2) and (l_3 ,m_3, n_3) , then the line having direction ratio l_1+l_2+l_3,m_1+ m_2+m_3, and n_1 + n_2 + n_3 , make an angle of

The direction cosines of the lines bisecting the angle between the line whose direction cosines are l_1, m_1, n_1 and l_2, m_2, n_2 and the angle between these lines is theta , are

The direction cosines of a line equally inclined to three mutually perpendiclar lines having direction cosines as l_(1),m_(1),n_(1),l_(2),m_(2),n_(2) and l_(3), m_(3),n_(3) are

If 1/2,1/3,n are direction cosines of a line, then the value of n is

O is the origin and lines OA, OB and OC have direction cosines l_(r),m_(r)andn_(r)(r=1,2 and3) . If lines OA', OB' and OC' bisect angles BOC, COA and AOB, respectively, prove that planes AOA', BOB' and COC' pass through the line (x)/(l_(1)+l_(2)+l_(3))=(y)/(m_(1)+m_(2)+m_(3))=(z)/(n_(1)+n_(2)+n_(3)) .

If l_1,""""m_1,""""n_1 and l_2,""""m_2,""""n_2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m_1n_2-m_2n_1, n_1l_2-n_2l_1, l_1m_2-l_2m_1 .

Let P M be the perpendicular from the point P(1,2,3) to the x-y plane. If vec O P makes an angle theta with the positive direction of the z- axis and vec O M makes an angle varphi with the positive direction of x- axis, w h e r eO is the origin and thetaa n dvarphi are acute angels, then a. costhetacosvarphi=1//sqrt(14) b. sinthetasinvarphi=2//sqrt(14) c. ""tanvarphi=2 d. tantheta=sqrt(5)//3

If I_n=int_0^(pi//4)tan^("n")x dx , prove that I_n+I_(n-2)=1/(n-1)dot

If I is the incenter of A B Ca n dR_1, R_2,a n dR_3 re, respectively, the radii of the circumcircles of the triangles I B C ,I C Aa n dI A B , then prove that R_1R_2R_3=2r R^2dot