Revealing the Mystery of Trigonometric Equations: A Proof of cosθ - sinθ √2sinθ

In the realm of trigonometry, certain identities can be both intriguing and challenging to prove. One such identity is the equation cos#x03B8; - sin#x03B8; 2sin#x03B8;. In this article, we will delve into the proof of this identity, exploring the underlying trigonometric relationships and identities that make it possible.

Proof of the Trigonometric Identity

Given the equation:

cos#x03B8; sin#x03B8; 2cos#x03B8;

We aim to prove that: cos#x03B8; - sin#x03B8; 2sin#x03B8;

Let's start with the given equation and manipulate it step by step:

cos#x03B8; sin#x03B8; 2cos#x03B8;

Multiply both sides by 2:

left21right>21right>#x03B8;sin#x03B8;2#x03B8;2

Expand and simplify:

left2cos#x03B8;sin#x03B8;2cos#x03B8;2right

left2cos#x03B8;2 2sin#x03B8;sin#x03B8;2cos#x03B8;2

left2cos#x03B8;2 sin#x03B8;22cos#x03B8;

Using the Pythagorean identity, cos#x03B8;2 sin#x03B8;21, we get:

leftcos#x03B8;2 sin#x03B8;22cos#x03B8;

Isolate cos#x03B8; - sin#x03B8;:

leftcos#x03B8;-sin#x03B8;2sin#x03B8;right

Thus, we have proven that cos#x03B8; - sin#x03B8; 2sin#x03B8;.

Verification of the Identity

It is important to note that for the equation cos#x03B8; - sin#x03B8; 2sin#x03B8; to hold true, the given equation cos#x03B8; sin#x03B8; 2cos#x03B8; must also hold true within the specified interval. This requires a specific value of #x03B8; that satisfies both conditions.

Additionally, it is worth noting that for trigonometric functions, the identity cos#x03B8; - sin#x03B8; 2sin#x03B8; doesn't hold true for all values of #x03B8;. It only holds for specific angles within the interval #x03B8; in (0, frac{pi}{4}) as derived from the monotonic properties of trigonometric functions.

Conclusion

The proof of the trigonometric identity cos#x03B8; - sin#x03B8; 2sin#x03B8; involves a series of algebraic manipulations and the use of trigonometric identities. It is a fascinating example of how trigonometric functions interrelate and how precise mathematical reasoning can uncover hidden relationships.