We have,
y=∣∣
∣∣sinxcosxsinxcosx−sinxcosxx11∣∣
∣∣
We know that,
if Δ=∣∣
∣
∣∣f1(x)f2(x)f3(x)g1(x)g2(x)g3(x)h1(x)h2(x)h3(x)∣∣
∣
∣∣
Then, Δ′=∣∣
∣
∣∣f′1(x)f′2(x)f′3(x)g1(x)g2(x)g3(x)h1(x)h2(x)h3(x)∣∣
∣
∣∣+∣∣
∣
∣∣f1(x)f2(x)f3(x)g′1(x)g′2(x)g′3(x)h1(x)h2(x)h3(x)∣∣
∣
∣∣+∣∣
∣
∣∣f1(x)f2(x)f3(x)g1(x)g2(x)g3(x)h′1(x)h′2(x)h′3(x)∣∣
∣
∣∣
∴dydx=∣∣
∣∣cosx−sinxcosxcosx−sinxcosxx11∣∣
∣∣+∣∣
∣∣sinxcosxsinx−sinx−cosx−sinxx11∣∣
∣∣+∣∣
∣∣sinxcosxsinxcosx−sinxcosx100∣∣
∣∣=0+(−1)⋅0+cos2x+sin2x=1