a) (cos^2 θ - sin^2 θ) / (cos^2 θ + sinθcosθ) = 1 - tanθ
L.H.S => (cos^2 θ - sin^2 θ)/{cosθ(cosθ + sinθ)}
=> {(cosθ + sinθ)(cosθ - sinθ)} / {cosθ(cosθ + sinθ)}
=> (cosθ - sinθ)/cosθ
=> 1 - tanθ = R.H.S
b) (sinθ + cosθ) (1+tan^2 θ) / (tanθ) = secθ + cscθ
L.H.S => (sinθ + cosθ) (1+tan^2 θ) /(tanθ)
=> (sinθ + cosθ) (sec^2{θ}) /(tanθ)
=> (sinθ + cosθ) (1/cos^2{θ}) /(tanθ)
=> (sinθ + cosθ) {1/(sinθcosθ) }
=> sinθ/(sinθcosθ) + cosθ/(sinθcosθ)
=> secθ + cscθ = R.H.S
c) (1 + tanθ) / (1 + cotθ) = (1 - tanθ) / (cotθ - 1)
L.H.S => (1 + tanθ) / (1 + cotθ)
=> {1 + sinθ/cosθ}/{1 + cosθ/sinθ}
=> {(cosθ + sinθ)/cosθ}/{(sinθ + cosθ)/sinθ
=> sinθ/cosθ
=> tanθ
R.H.S => (1 - tanθ) / (cotθ - 1)
=> {1 - sinθ/cosθ}/{cosθ/sinθ - 1}
=> {(cosθ - sinθ)/cosθ}/{(cosθ - sinθ)/sinθ}
=> sinθ/cosθ
=> tanθ
Hence, L.H.S = R.H.S
d) sin^2 θ - sin^6 θ + cos^2 θ - cos^6 θ = 3sin^2 θcos^2 θ
rearranging terms in L.H.S
=> sin^2 θ + cos^2 θ - sin^6 θ - cos^6 θ
=> 1 - (sin^6 θ - cos^6 θ)
=> 1 - { (sin^2 θ)^3 + (cos^2 θ)^3}
Applying a^3 + b^3 = (a+b)^3 − 3ab(a+b)]
=> 1 - { (sin^2 θ + cos^2 θ)^3 - 3sin^2 θ * cos^2 θ(sin^2 θ+ cos^2 θ)}
=> 1 - { 1 - 3sin^2 θ * cos^2 θ(1)}
=> 3sin^2 θ * cos^2 θ = R.H.S