If `3cos alpha-4sinalpha=5`, show that, `3sinalpha+4cosalpha=0` .

If cosalpha+secalpha=2 , show that cos^3alpha+sec^3alpha=2 .

if tan theta=(sinalpha-cosalpha)/(sinalpha+cosalpha) show that, sin alpha+cosalpha=pmsqrt(2)cos theta

If A =[[cos alpha,-sin alpha],[sinalpha, cos alpha]] , B= [[cos 2 beta, sin 2 beta],[sin beta ,-cos beta]] where 0 lt beta lt pi/2 , then prove that BAB=A^(-1) . Also, find the least value of alpha of which BA^(4) B= A^(-1)

If f(alpha)=[[cosalpha,-sinalpha,0],[sinalpha,cosalpha,0],[ 0, 0, 1]],t h e n prove that [F(alpha)]^(-1)=F(-alpha)dot

If (tan3alpha)/(tanalpha)=lambda ,show that (sin3alpha)/(sinalpha)=(2lambda)/(lambda-1) and hence prove that the value of lambda does not lie between 1/3 and 3

If cosx + sinx = cos alpha- sinalpha , prove that, x-pi/4 = 2npi +-(alpha + pi/4)

Prove that, 1/3 ( cos^(3) alpha sin 3 alpha + sin^(3) alpha cos 3 alpha) = 1/4 sin 4 alpha

If A(alpha,beta)=[[cosalpha,sinalpha,0],[-sinalpha,cosalpha,0],[ 0, 0,e^(beta)]],t h e nA(alpha,beta)^(-1) is equal to a. A(-alpha,-beta) b. A(-alpha,beta) c. A(alpha,-beta) d. A(alpha,beta)

Prove that the product of the matrices {:[(cos ^(2)alpha,cos alpha sin alpha ),(cos alpha sinalpha, sin^(2)alpha)]and {:[(cos ^(2)beta,cosbetasinbeta),(cos betasinbeta,sin^(2)beta)] is the null matrix when alpha and beta differ by an odd multiple of (pi)/(2) .

If A={:[(0,-tan.alpha/2),(tan.alpha/2,0)]:} and I is the identify matrix of order 2, Show that I+A=(I-A) {:[(cosalpha,-sinalpha),(sinalpha,cosalpha)]:} .