Prove that `sin^5theta=1/16[sin5theta-5sin3theta+10sintheta]`

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

Expand `2x^3+7x^2+x-1` in powers of x - 2

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

Find all values of `(1 + i)^(1/3` and show that their continued product is (1+ 𝒊 ).

Concept: Powers and Roots of Trigonometric Functions

By using De Moivre's Theorem obtain tan 5θ in terms of tan θ and show that `1-10 tan^2(pi/10)+5tan^4(pi/10)=0`.

Concept: D’Moivre’S Theorem

If y=(x+√x^{2}-1 ,Prove that

`(x^2-1)y_(n+2)+(2n+1)xy_(n+1)+(n^2-m^2)y_n=0`

Concept: Leibnitz’S Theorem (Without Proof) and Problems

Find the n^th derivative of `x^3/((x+1)(x-2))`

Concept: nth derivative of standard functions

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

Concept: nth derivative of standard functions

If `y=e^(tan^(-1)x)`.Prove that

`(1+x^2)y_(n+2)+[2(n+1)x-1]y_(n+1)+n(n+1)y_n=0`

Concept: Leibnitz’S Theorem (Without Proof) and Problems

Evaluate `lim_(x->0) sinx log x.`

Concept: nth derivative of standard functions

If U = `e^(xyz) f((xy)/z)` prove that `x(delu)/(delx)+z(delu)/(delx)2xyzu` and `y(delu)/(delx)+z(delu)/(delz)=2xyzu` and hence show that `x(del^2u)/(delzdelx)=y(del^2u)/(delzdely)`

Concept: Successive Differentiation

If y=sin[log(x^{2}+2x+1)] then prove that (x+1)^{2}y_{n+2} +(2n +1)(x+ 1)y_{n+1} + (n^{2}+4)y_{n}=0.

Concept: Leibnitz’S Theorem (Without Proof) and Problems

Find nth derivative of `1/(x^2+a^2.`

Concept: nth derivative of standard functions

Prove that log `[tan(pi/4+(ix)/2)]=i.tan^-1(sinhx)`

Concept: Logarithmic Functions

Obtain tan 5𝜽 in terms of tan 𝜽 & show that `1-10tan^2 x/10+5tan^4 x/10=0`

Concept: Separation of Real and Imaginary Parts of Logarithmic Functions

If y=etan_1x. prove that `(1+x^2)yn+2[2(n+1)x-1]y_n+1+n(n+1)y_n=0`

Concept: Separation of Real and Imaginary Parts of Logarithmic Functions

If `Z=x^2 tan-1y /x-y^2 tan -1 x/y del`

Prove that `(del^z z)/(del_ydel_x)=(x^2-y^2)/(x^2+y^2)`

Concept: Logarithmic Functions

Find tanhx if 5sinhx-coshx = 5

Concept: Separation of Real and Imaginary Parts of Logarithmic Functions

Separate into real and imaginary parts of cos`"^-1((3i)/4)`

Concept: Separation of Real and Imaginary Parts of Logarithmic Functions

Considering only principal values separate into real and imaginary parts

`i^((log)(i+1))`

Concept: Separation of Real and Imaginary Parts of Logarithmic Functions

Show that `ilog((x-i)/(x+i))=pi-2tan6-1x`

Concept: Logarithmic Functions

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