Substitute the P_l(x), you found in Problems 4.3 or 5.3into equation (10.6)to find P_l^m(x) then let x=cosθ to evaluate:

P₁¹(cosθ)

An equation similar to Legendre’s equation is-

(1-x²) y"-2xy'+[l(l+1)-m²/1-x²] y=0 (∀ m²≤ l²)

The solution to this equation is of the form

P_l^m (x)=(1-x²)^m/2 d^m/dx^m (P_l(x))

These solutions are called Associated Legendre Functions.

It is given that

P_l^m(x) =(1-x²)^m/2 d^m/dx^mP_l(x) …… (1)

Substitute l=1 and m=1 to obtain,

P₁¹(x) = (1-x²)^1/2 d¹/dx¹ P₁(x)

P₁¹(x) = (1-x²)^1/2 d/dx P₁(x)

P₁¹(x) = (1-x²)^1/2 d/dx (x)

P₁¹(x) = (1-x²)^1/2 (1)

Therefore,

P₁¹(x) = (1-x²)^1/2 …… (2)

On substituting x=cos x into equation (2):

P₁¹(x)=(1-x²)^1/2

P₁¹(cos θ)= (1-cos²θ)^1/2

P₁¹(cos θ)= (sin²θ)^1/2

P₁¹(cos θ)= sinθ

Hence, the required value is P₁¹(cos θ)= sinθ.