To find the expression for [tex]\( y = \sin^2(x) - \tan^2(x) \)[/tex], let's break down each term and understand the involved trigonometric functions in detail.
1. Understand [tex]\(\sin^2(x)\)[/tex]:
- [tex]\(\sin(x)\)[/tex] is the sine of angle [tex]\(x\)[/tex].
- [tex]\(\sin^2(x)\)[/tex] means [tex]\((\sin(x))^2\)[/tex], which is the square of the sine of angle [tex]\(x\)[/tex].
2. Understand [tex]\(\tan^2(x)\)[/tex]:
- [tex]\(\tan(x)\)[/tex] is the tangent of angle [tex]\(x\)[/tex], which can also be expressed as [tex]\(\frac{\sin(x)}{\cos(x)}\)[/tex].
- [tex]\(\tan^2(x)\)[/tex] means [tex]\((\tan(x))^2\)[/tex], which is the square of the tangent of angle [tex]\(x\)[/tex], and can be written as [tex]\(\left(\frac{\sin(x)}{\cos(x)}\right)^2\)[/tex], or [tex]\(\frac{\sin^2(x)}{\cos^2(x)}\)[/tex].
3. Combine the two terms:
- We need to subtract [tex]\(\tan^2(x)\)[/tex] from [tex]\(\sin^2(x)\)[/tex].
Let's express the entire expression step by step:
[tex]\[ y = \sin^2(x) - \tan^2(x) \][/tex]
Since [tex]\(\tan(x) = \frac{\sin(x)}{\cos(x)}\)[/tex], so [tex]\(\tan^2(x) = \left(\frac{\sin(x)}{\cos(x)}\right)^2 = \frac{\sin^2(x)}{\cos^2(x)}\)[/tex].
Substitute [tex]\(\tan^2(x)\)[/tex] in the expression:
[tex]\[ y = \sin^2(x) - \frac{\sin^2(x)}{\cos^2(x)} \][/tex]
We notice that we can factor out [tex]\(\sin^2(x)\)[/tex] from both terms in the expression:
[tex]\[ y = \sin^2(x) \left(1 - \frac{1}{\cos^2(x)}\right) \][/tex]
Now, remember that [tex]\(\frac{1}{\cos^2(x)}\)[/tex] is [tex]\(\sec^2(x)\)[/tex]:
[tex]\[ y = \sin^2(x) \left(1 - \sec^2(x)\right) \][/tex]
Using the trigonometric identity [tex]\(\sec^2(x) = 1 + \tan^2(x)\)[/tex], we get:
[tex]\[ y = \sin^2(x) \left(1 - (1 + \tan^2(x))\right) \][/tex]
[tex]\[ y = \sin^2(x) \left(1 - 1 - \tan^2(x)\right) \][/tex]
[tex]\[ y = \sin^2(x) \left(-\tan^2(x)\right) \][/tex]
Finally, we know from trigonometric identities that:
[tex]\[ y = \sin(x)^2 - \tan(x)^2 \][/tex]
So, the correctly simplified expression for the function [tex]\(y\)[/tex] in terms of trigonometric functions is:
[tex]\[ y = \sin^2(x) - \tan^2(x) \][/tex]
Hence, the detailed expression for [tex]\( y \)[/tex] is:
[tex]\[ y = \sin^2(x) - \tan^2(x) \][/tex]
This completes our derivation and we have confirmed that the expression for [tex]\( y \)[/tex] is [tex]\( \sin^2(x) - \tan^2(x) \)[/tex].
