Understanding the Trigonometry Problem

The question asks us to find the value of the expression $\text{cosec}^2\theta – \sec^2\theta$, given that $\tan \theta = 1/\sqrt{5}$. To solve this, we need to determine the values of $\text{cosec}^2\theta$ and $\sec^2\theta$ using the given value of $\tan \theta$. We can use fundamental trigonometric identities to relate $\tan \theta$ to $\sec^2\theta$ and $\text{cosec}^2\theta$.

Finding sec²θ and cosec²θ from tan θ

We know two key Pythagorean identities:
1. $\sec^2\theta = 1 + \tan^2\theta$
2. $\text{cosec}^2\theta = 1 + \cot^2\theta$

Since we are given $\tan \theta$, we can directly find $\sec^2\theta$ using the first identity. To find $\text{cosec}^2\theta$, we first need to find $\cot \theta$. We know that $\cot \theta$ is the reciprocal of $\tan \theta$, i.e., $\cot \theta = 1/\tan \theta$. Once we have $\cot \theta$, we can find $\text{cosec}^2\theta$ using the second identity.

Calculating sec²θ using tan θ

Given $\tan \theta = 1/\sqrt{5}$. Using the identity $\sec^2\theta = 1 + \tan^2\theta$:

Substitute the given value of $\tan \theta$:

To add these values, find a common denominator:

So, the value of $\sec^2\theta$ is $6/5$.

Calculating cosec²θ using tan θ

First, find $\cot \theta$ using the reciprocal identity $\cot \theta = 1/\tan \theta$.

Substitute the given value of $\tan \theta = 1/\sqrt{5}$:

Now, use the identity $\text{cosec}^2\theta = 1 + \cot^2\theta$:

Substitute the value of $\cot \theta = \sqrt{5}$:

So, the value of $\text{cosec}^2\theta$ is $6$.

Calculating cosec²θ – sec²θ

Now that we have the values for $\text{cosec}^2\theta$ and $\sec^2\theta$, we can find the value of the expression $\text{cosec}^2\theta – \sec^2\theta$.

We found $\text{cosec}^2\theta = 6$ and $\sec^2\theta = 6/5$.

To subtract these values, find a common denominator:

The value of $\text{cosec}^2\theta – \sec^2\theta$ is $24/5$.

Final Answer Value

The calculated value of cosec²θ – sec²θ is 24/5.

Revision Table: Essential Trigonometric Identities

Additional Information: Relationship Between Trigonometric Ratios

Trigonometric ratios define the relationship between the angles and sides of a right-angled triangle. The six basic ratios are sine, cosine, tangent, cosecant, secant, and cotangent. These ratios are interconnected through various identities. Understanding these identities is crucial for simplifying trigonometric expressions and solving equations.

For instance, the Pythagorean identities are derived from the Pythagorean theorem ($a^2 + b^2 = c^2$). If you consider a right-angled triangle with angle $\theta$, opposite side $a$, adjacent side $b$, and hypotenuse $c$, then $\sin\theta = a/c$, $\cos\theta = b/c$, and $\tan\theta = a/b$. Dividing $a^2 + b^2 = c^2$ by $c^2$ gives $(a/c)^2 + (b/c)^2 = (c/c)^2$, which simplifies to $\sin^2\theta + \cos^2\theta = 1$. Similarly, dividing by $b^2$ or $a^2$ leads to the other Pythagorean identities involving $\sec^2\theta$ and $\text{cosec}^2\theta$. These relationships allow us to find any trigonometric ratio if one ratio is known for a particular angle, provided the angle is in a suitable domain.