Solving \sin^3\theta + \operatorname{cosec}^3\theta Value from \sin\theta + \operatorname{cosec}\theta = \sqrt{5}

The problem asks us to find the value of the expression $\sin^3\theta + \operatorname{cosec}^3\theta$ given the condition that $\sin \theta + \operatorname{cosec} \theta = \sqrt{5}$. This can be solved efficiently using algebraic identities combined with trigonometric properties.

Understanding the Trigonometric Relationship

We are provided with the equation:

$$ \sin \theta + \operatorname{cosec} \theta = \sqrt{5} $$

Our goal is to determine the value of $\sin^3\theta + \operatorname{cosec}^3\theta$.

First, let's recall the fundamental relationship between cosecant and sine:

$$ \operatorname{cosec}\theta = \frac{1}{\sin\theta} $$

Using this, we can find the product of $\sin\theta$ and $\operatorname{cosec}\theta$:

$$ \sin\theta \times \operatorname{cosec}\theta = \sin\theta \times \frac{1}{\sin\theta} $$

Assuming $\sin\theta \neq 0$, this simplifies to:

$$ \sin\theta \times \operatorname{cosec}\theta = 1 $$

Applying the Sum of Cubes Identity

We can use the algebraic identity for the sum of cubes, which is $a^3 + b^3 = (a+b)^3 - 3ab(a+b)$.

Let's set $a = \sin\theta$ and $b = \operatorname{cosec}\theta$. Substituting these into the identity gives us:

$$ \sin^3\theta + \operatorname{cosec}^3\theta = (\sin\theta + \operatorname{cosec}\theta)^3 - 3(\sin\theta)(\operatorname{cosec}\theta)(\sin\theta + \operatorname{cosec}\theta) $$

Step-by-Step Calculation

Now, we substitute the values we know into the derived formula:

• From the problem statement: $\sin\theta + \operatorname{cosec}\theta = \sqrt{5}$
• From our calculation: $\sin\theta \times \operatorname{cosec}\theta = 1$

Substituting these into the equation:

$$ \sin^3\theta + \operatorname{cosec}^3\theta = (\sqrt{5})^3 - 3(1)(\sqrt{5}) $$

Let's evaluate the terms:

• The term $(\sqrt{5})^3$ is $\sqrt{5} \times \sqrt{5} \times \sqrt{5}$, which equals $5\sqrt{5}$.
• The term $3(1)(\sqrt{5})$ is simply $3\sqrt{5}$.

Now, perform the subtraction:

$$ \sin^3\theta + \operatorname{cosec}^3\theta = 5\sqrt{5} - 3\sqrt{5} $$

$$ \sin^3\theta + \operatorname{cosec}^3\theta = (5 - 3)\sqrt{5} $$

$$ \sin^3\theta + \operatorname{cosec}^3\theta = 2\sqrt{5} $$

Conclusion

Therefore, the value of the expression $\sin^3\theta + \operatorname{cosec}^3\theta$ is $2\sqrt{5}$.