If xϵR and nϵI, then the determinant Δ=∣∣ ∣ ∣∣sin(nπ)sinx−cosxlog(tanx)cosx−sinxcos[(2n+1)π2]log(cotx)log(cotx)log(tanx)tan(nπ)∣∣ ∣ ∣∣
0
We can write Δ as
Δ=∣∣ ∣ ∣∣0sinx−cosxlog(tanx)−(sinx−cosx)0−log(tanx)−log(tanx)log(tanx)0∣∣ ∣ ∣∣
=(−1)3∣∣ ∣ ∣∣0−(sinx−cosx)−log(tanx)sinx−cosx0log(tanx)log(tanx)−log(tanx)0∣∣ ∣ ∣∣ [taking −1 common from R1, R2 and R3]
=−Δ
⇒2Δ=0⇒Δ=0