3(i)
Factorize: sin3 θ + cos3 θ
Hence, prove the following identity :
$\frac{\text{sin}^3 θ + \text{cos}^3 θ}{\text{sin θ + cos θ}}$ + sin θ cos θ = 1
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Factorizing,
⇒ sin3 θ + cos3 θ = (sin θ + cos θ)(sin2 θ + cos2 θ - sin θ cos θ)
⇒ sin3 θ + cos3 θ = (sin θ + cos θ)(1 - sin θ cos θ) ...........(1)
To prove,
$\frac{\text{sin}^3 θ + \text{cos}^3 θ}{\text{sin θ + cos θ}}$ + sin θ cos θ = 1
Substituting value of sin3 θ + cos3 θ from equation (1) in L.H.S. of above equation :
⇒ $\frac{\text{(sin θ + cos θ)(1 - sin θ cos θ)}}{\text{sin θ + cos θ}} + \text{sin θ cos θ}$
⇒ 1 - sin θ cos θ + sin θ cos θ
⇒ 1.
Since, L.H.S. = R.H.S.
Hence, proved that $\frac{\text{sin}^3 θ + \text{cos}^3 θ}{\text{sin θ + cos θ}} + \text{sin θ cos θ} = 1$.