QQuestion
Factorize: sin³ θ + cos³ θ.
Hence, prove the identity:
\frac{\sin^3\theta+\cos^3\theta}{\sin\theta+\cos\theta}+\sin\theta\cos\theta=1✓Answer
Solution:
Using a³ + b³ = (a + b)(a² − ab + b²):
sin³ θ + cos³ θ
= (sin θ + cos θ)(sin² θ − sin θ cos θ + cos² θ)
Since sin² θ + cos² θ = 1:
sin³ θ + cos³ θ = (sin θ + cos θ)(1 − sin θ cos θ) … (1)
Now consider the left-hand side of the identity:
\frac{\sin^3\theta+\cos^3\theta}{\sin\theta+\cos\theta}+\sin\theta\cos\thetaSubstituting from (1):
\frac{(\sin\theta+\cos\theta)(1-\sin\theta\cos\theta)}{\sin\theta+\cos\theta}+\sin\theta\cos\theta= 1 − sin θ cos θ + sin θ cos θ
= 1
Therefore, LHS = RHS. Hence proved.
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