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ICSEClass 10MathematicsTrigonometrical Identities03 Mark2024 Moderate

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

Exam-ready answer • 03 Mark

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\theta

Substituting 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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