Question

If \(\sec x=P, \operatorname{cosec} x=Q\), then : (a) \(P^{2}+Q^{2}=P Q\) (b) \(P^{2}+Q^{2}=P^{2} Q^{2}\) (c) \(P^{2}-Q^{2}=P^{2} Q^{2}\) (d) \(P^{2}+Q^{2}=-P^{2} Q^{2}\)

Short answer

(a) \(P^2 + Q^2 = PQ\) (b) \(P^2 + Q^2 = P^2Q^2\) (c) \(P^2 - Q^2 = P^2Q^2\) (d) \(P^2 + Q^2 = -P^2Q^2\) Answer: None of the given options are correct. The derived equation, \((P^2-1)(Q^2)=1\), does not match any of the given options.

Step-by-step solution

Remember the definitions of sec(x) and cosec(x)

Recall that the secant function, \(\sec x = \frac{1}{\cos x}\), and the cosecant function, \(\operatorname{cosec} x = \frac{1}{\sin x}\).

Use the Pythagorean trigonometric identity

We know the identity: \(\sin^2 x + \cos^2 x = 1\). Now, we will write this identity in terms of \(\sec x\) and \(\operatorname{cosec} x\). \(1-\cos^2 x = \sin^2 x => \frac{1-\cos^2 x}{\cos^2 x} = \frac{\sin^2 x}{\cos^2 x}\). Since \(\sec x = \frac{1}{\cos x}\) and \(\operatorname{cosec} x = \frac{1}{\sin x}\), we can rewrite the above equation as: \((\sec^2 x -1)(\operatorname{cosec}^2 x) = 1\)

Test each option using the obtained equation

Substitute \(P = \sec x\) and \(Q = \operatorname{cosec} x\) in the above equation: \((P^2-1)(Q^2)=1\) Now, let's check each option: (a) \(P^2 + Q^2 = PQ\) There's no direct way to rewrite the equation \((P^2-1)(Q^2)=1\) to match this option. So, option (a) is not correct. (b) \(P^2 + Q^2 = P^2Q^2\) Rewrite the equation \((P^2-1)(Q^2)=1\) as \(P^2Q^2 - Q^2 = 1\). This equation doesn't match the given option. So, option (b) is not correct. (c) \(P^2 - Q^2 = P^2Q^2\) This option cannot be derived from the equation \((P^2-1)(Q^2)=1\). So, option (c) is not correct. (d) \(P^2 + Q^2 = -P^2Q^2\) We can't rewrite the equation \((P^2-1)(Q^2)=1\) to match this option. So, option (d) is not correct. It turns out that none of the given options are correct. The equation derived from the trigonometric identity does not match any of the given options.

Key concepts

Secant Function

The secant function plays a key role in trigonometry, although it doesn't usually get the same spotlight as its more commonly discussed counterparts like sine and cosine. Its definition as the reciprocal of the cosine function is essential.

Secant is expressed as \( \sec x = \frac{1}{\cos x} \), where x is an angle. This means that for any angle, x, secant is the ratio of the length of the hypotenuse to the adjacent side in a right-angled triangle.

When solving problems involving secant, you must be aware that because it's the flip of cosine, whenever cosine is zero, secant is undefined, since you cannot divide by zero. This occurs at x values where \( \cos x = 0 \), specifically at \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \).

In the given exercise and solution, understanding secant was pivotal to deriving the correct equation. The step of identifying that \( \sec^2 x - 1 = \tan^2 x \), is a rearrangement of the Pythagorean identity, forms the groundwork of knowledge needed to tackle trigonometric problems efficaciously.

Cosecant Function

On the other side, we find the cosecant function, which is similarly overshadowed by its more primary partner, sine. The definition of cosecant mirrors that of secant but in relation to the sine function: \( \operatorname{cosec} x = \frac{1}{\sin x} \), where x is also an angle.

The cosecant function, therefore, represents the ratio of the hypotenuse's length to the opposite side in a right-angled triangle. Similar to secant, it is undefined when the sine of an angle is zero — occurring at whole number multiples of \( \pi \)

Having a strong grasp of the cosecant function was necessary for the exercise. By recognizing its relationship with sine and making use of the identity \( \operatorname{cosec}^2 x = 1 + \cot^2 x \), you would understand how it interacts within equations and why none of the options given in the exercise matched after transforming the Pythagorean identity.

Pythagorean Trigonometric Identity

Central to the study of trigonometry is the Pythagorean trigonometric identity, which states that for any angle x: \( \sin^2 x + \cos^2 x = 1 \). It's derived from the Pythagorean Theorem, applied to a unit circle where the radius (hypotenuse) is 1.

This fundamental relationship between sine and cosine provides a foundational understanding from which many other identities are derived, including the expressions for secant and cosecant functions. In practical terms, this identity shows the intrinsic link between the two basic trigonometric functions, no matter the angle.

During the exercise, the Pythagorean identity was crucial in determining the relationship between the values of P and Q where P represented secant and Q represented cosecant. The manipulation and understanding of this identity showed that none of the proposed options were correct. To avoid confusion for students, highlighting how each trigonometric identity is applied and transformed is essential for a clear and accurate understanding.