Show that `((x+b)(x+c))/((b-a)(c-a))+((x+c)(x+a))/((c-b)(a-b))+((x+a)(x+b))/((a-c)(b-c))=1` is an identity.

Prove that x^((b-c)/(bc)) x^((c-a)/(ca)) x^((a-b)/(ab))=1

Prove: x^(log(b/c) ) x^(log((c)/(a))) x^(log((a)/(b)))=1

Solve for x in R : |((x+a)(x-a),(x+b)(x-b),(x+c)(x-c)),((x-a)^3,(x-b)^3,(x-c)^3),((x+a)^3,(x+b)^3,(x+c)^3)| =0, a,b and c being distinct real numbers.

Prove that |((b+x)(c+x),(c+x)(a+x),(a+x)(b+x)),((b+y)(c+y),(c+y)(a+y),(a+y)(b+y)),((b+z)(c+z),(c+z)(a+z),(a+z)(b+z))|=(a-b)(b-c)(c-a)(x-y)(y-z)(z-x)dot

Simplify (x^(a-b))^(a+b).(x^(b-c))^(b+c).(x^(c-a))^(c+a)

Prove that |(b+x)(c+x)(v+x)(a+x)(a+x)(b+x)(b+y)(c+y)(c+x)(a+t)(a+y)(b+y)(b+z)(c+z)(c+z)(a+z)(a+z)(b+z)|=(b-c)(c-a)(y-z)(x-y)dot

Prove that the minimum value of ((a+x)(b+x))/((c+x))a ,b > c ,x >-c is (sqrt(a-c)+sqrt(b-c))^2

(a) If x + y + z=0, show that x ^(3) + y ^(3) + z ^(3)= 3 xyz. (b) Show that (a-b) ^(3) + (b-c) ^(3) + (c-a)^(3) =3 (a-b) (b-c) (c-a)

Show that |(x+2a,y+2b,z+2c),(x,y,z),(a,b,c)|=0.

If y=f(x) is a monotonic function in (a,b), then the area bounded by the ordinates at x=a, x=b, y=f(x) and y=f(c)("where "c in (a,b))" is minimum when "c=(a+b)/(2) . "Proof : " A=int_(a)^(c)(f(c)-f(x))dx+int_(c)^(b)(f(c))dx =f(c)(c-a)-int_(a)^(c) (f(x))dx+int_(a)^(b)(f(x))dx-f(c)(b-c) rArr" "A=[2c-(a+b)]f(c)+int_(c)^(b)(f(x))dx-int_(a)^(c)(f(x))dx Differentiating w.r.t. c, we get (dA)/(dc)=[2c-(a+b)]f'(c)+2f(c)+0-f(c)-(f(c)-0) For maxima and minima , (dA)/(dc)=0 rArr" "f'(c)[2c-(a+b)]=0(as f'(c)ne 0) Hence, c=(a+b)/(2) "Also for "clt(a+b)/(2),(dA)/(dc)lt0" and for "cgt(a+b)/(2),(dA)/(dc)gt0 Hence, A is minimum when c=(a+b)/(2) . If the area enclosed by f(x)= sin x + cos x, y=a between two consecutive points of extremum is minimum, then the value of a is