`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`

Find the angle between the line whose direction cosines are given by l+m+n=0a n dl^2+m^2-n^2=0.

Find the angle between the lines whose direction cosines are connected by the relations l+m+n=0a n d2lm+2n l-m n=0.

A line passes through the points (6,-7,-1)a n d(2,-3,1)dot Find te direction cosines off the line if the line makes an acute angle with the positive direction of the x-axis.

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)) .

The pair of lines whose direction cosines are given by the equations 3l+m+5n=0a n d6m n-2n l+5l m=0 are a. parallel b. perpendicular c. inclined at cos^(-1)(1/6) d. none of these

Let A B C be a triangle with incenter I and inradius rdot Let D ,E ,a n dF be the feet of the perpendiculars from I to the sides B C ,C A ,a n dA B , respectively. If r_1,r_2a n dr_3 are the radii of circles inscribed in the quadrilaterals A F I E ,B D I F ,a n dC E I D , respectively, prove that (r_1)/(r-1_1)+(r_2)/(r-r_2)+(r_3)/(r-r_3)=(r_1r_2r_3)/((r-r_1)(r-r_2)(r-r_3))

Prove that (C_1)/2+(C_3)/4+(C_5)/6+=(2^(n)-1)/(n+1)dot

If t a n^2theta=2t a n^2varphi+1 , prove that cos2theta+sin^2phi =0.

If a > b and n is a positive integer, then prove that a^n-b^n > n(a b)^((n-1)//2)(a-b)dot

Consider two lines L_1a n dL_2 given by a_1x+b_1y+c_1=0a n da_2x+b_2y+c_2=0 respectively where c1 and c2 !=0, intersecting at point PdotA line L_3 is drawn through the origin meeting the lines L_1a n dL_2 at Aa n dB , respectively, such that PA=P B . Similarly, one more line L_4 is drawn through the origin meeting the lines L_1a n dL_2 at A_1a n dB_2, respectively, such that P A_1=P B_1dot Obtain the combined equation of lines L_3a n dL_4dot