Suppose `alpha,beta,gammaa n ddelta` are the interior angles of regular pentagon, hexagon, decagon, and dodecagon, respectively, then the value of `|cosalphasecbetacosgammacos e cdelta|` is _________

If a line makes angles alpha,betaa n dgamma with three-dimensional coordinate axes, respectively, then find the value of cos2alpha+cos2beta+cos2gammadot

If alpha,beta,a n dgamma are the angles which a directed line makes with the positive directions of the co-ordinates axes, then find the value of sin^2alpha+sin^2beta+sin^2gammadot

If vec alpha, vec beta, vecgamma be three unit vectors such that |vec alpha+vec beta+vec gamma|=1 and vec alpha is perpendicular to vec beta while vec gamma makes angles theta and phi with vec alpha and vec beta respectively, then the value of cos theta+cos phi is -

If alpha,beta,gamma,delta are the roots of the equation x^4-K x^3+K x^2+L x+m=0,w h e r eK ,L ,a n dM are real numbers, then the minimum value of alpha^2+beta^2+gamma^2+delta^2 is a. 0 b. -1 c. 1 d. 2

A line makes angles alpha,beta,gammaa n ddelta with the diagonals of a cube. Show that cos^2alpha+cos^2beta+cos^2gamma+cos^2delta=4//3.

vec u , vec v and vec w are three non-coplanar unit vecrtors and alpha,beta and gamma are the angles between vec u and vec v , vec v and vec w ,a n d vec w and vec u , respectively, and vec x , vec y and vec z are unit vectors along the bisectors of the angles alpha,betaa n dgamma , respectively. Prove that [ vecx xx vec y vec yxx vec z vec zxx vec x]=1/(16)[ vec u vec v vec w]^2sec^2(alpha/2)sec^2(beta/2)sec^2(gamma/2) .

Suppose alpha,beta a n d theta are angles satisfying 0 < alpha < theta < beta < pi/2 dot Then prove that (sin alpha-sin beta)/(cos beta-cos alpha)=-cot theta

Three parallel chords of a circle have lengths 2,3,4 units and subtend angles alpha,beta,alpha+beta at the centre, respectively (alphaltbetaltpi), then find the value of cosalpha

Computing area with parametrically represented boundaries : If the boundary of a figure is represented by parametric equation, i.e., x=x(t), y=(t), then the area of the figure is evaluated by one of the three formulas : S=-int_(alpha)^(beta)y(t)x'(t)dt, S=int_(alpha)^(beta)x(t)y'(t)dt, S=(1)/(2)int_(alpha)^(beta)(xy'-yx')dt, Where alpha and beta are the values of the parameter t corresponding respectively to the beginning and the end of the traversal of the curve corresponding to increasing t. The area of the loop described as x=(t)/(3)(6-t),y=(t^(2))/(8)(6-t) is

If alpha,a n dbeta be the roots of the equation x^2+p x-1//2p^2=0,w h e r ep in Rdot Then the minimum value of alpha^4+beta^4 is 2sqrt(2) b. 2-sqrt(2) c. 2 d. 2+sqrt(2)