`"If" x=uv & y=u/v "prove that" jj^1=1`

Concept: .Circular Functions of Complex Number

Prove that `sin^(-1)(cosec theta)=pi/2+i.log(cot theta/2)`

Concept: Expansion of sinn θ, cosn θ in terms of sines and cosines of multiples of θ

Prove that 𝒕𝒂𝒏𝒉−𝟏(𝒔𝒊𝒏 𝜽) = 𝒄𝒐𝒔𝒉−𝟏(𝒔𝒆𝒄 𝜽)

Concept: .Circular Functions of Complex Number

If `cos alpha cos beta=x/2, sinalpha sinbeta=y/2`, prove that:

`sec(alpha -ibeta)+sec(alpha-ibeta)=(4x)/(x^2-y^2)`

Concept: Review of Complex Numbers‐Algebra of Complex Number

Prove that the matrix `1/sqrt3` `[[ 1,1+i1],[1-i,-1]]` is unitary.

Concept: .Circular Functions of Complex Number

If `z =log(e^x+e^y) "show that rt" - s^2 = 0 "where r"= (del^2z)/(delx^2),t=(del^2z)/(dely^2)"s"=(del^2z)/(delx dely)`

Concept: Review of Complex Numbers‐Algebra of Complex Number

If Z=tan^1 (x/y), where` x=2t, y=1-t^2, "prove that" d_z/d_t=2/(1+t^2).`

Concept: Review of Complex Numbers‐Algebra of Complex Number

Find the nth derivative of cos 5x.cos 3x.cos x.

Concept: Review of Complex Numbers‐Algebra of Complex Number

Evaluate : `Lim_(x→0) (x)^(1/(1-x))`

Concept: Review of Complex Numbers‐Algebra of Complex Number

If x = uv, y `=(u+v)/(u-v).`find `(del(u,v))/(del(x,y))`.

Concept: Review of Complex Numbers‐Algebra of Complex Number

Find all values of `(1+i)^(1/3)` & show that their continued

Product is (1+i).

Concept: D’Moivre’S Theorem

If `y=2^xsin^2x cosx` find `y_n`

Concept: Review of Complex Numbers‐Algebra of Complex Number

If u=`f((y-x)/(xy),(z-x)/(xz)),"show that" x^2 (del_u)/(del_x)+y^2 (del_u)/(del_y)+x^2 del_u/del_z=0`

Concept: .Circular Functions of Complex Number

Show that the roots of x^{5 }=1 can be written as 1, `alpha^1,alpha^2,alpha^3,alpha^4` .hence show that `(1-alpha^1) (1-alpha^2) (1-alpha^3)(1-alpha^4)=5.`

Concept: Powers and Roots of Trigonometric Functions

If `u=x^2+y^2+z^2` where `x=e^t, y=e^tsint,z=e^tcost`

Prove that `(du)/(dt)=4e^(2t)`

Concept: Review of Complex Numbers‐Algebra of Complex Number

Expand `2x^3+7x^2+x-6` in powers of (x-2)

Concept: Expansion of sinnθ, cosnθ in powers of sinθ, cosθ

Using De Moivre’s theorem prove that]

`cos^6theta-sin^6theta=1/16(cos6theta+15cos2theta)`

Concept: D’Moivre’S Theorem

If `tan(x/2)=tanh(u/2),"show that" u = log[(tan(pi/4+x/2))] `

Concept: Inverse Hyperbolic Functions

Show that all roots of `(x+1)^6+(x-1)^6=0` are given by -icot`((2k+1)n)/12`where k=0,1,2,3,4,5.

Concept: Powers and Roots of Trigonometric Functions