Question

if tan A=√2-1 show that sin A cos A=√2/4​

Answer 1

tanA=[2]^(1/2)-1

sinA/cosA=[2]^(1/2)-1

multiply numerator and denominator by cosA

sinAcosA/cos^2A=[2]^(1/2)-1

sinAcosAsec^2A=[2]^(1/2)-1

sinAcosA(1+tan^2A)=[2]^(1/2)-1

sinAcosA(4-2[2]^(1/2))=[2]^(1/2)-1

sinAcosA=[2]^(1/2)-1/4-2[2]^(/2)

rationalise,

sinAcosA={[2]^(1/2)-1}{4+2[2]^(1/2)}/8

=2[2]^(1/2)/8=[2]^(1/2)/4

we can also prove it using a right angled triangle