Proving the Trigonometric Identity: √(1 - sinx) cos(x/2) - sin(x/2)

Introduction:

In trigonometry, the relationship between trigonometric functions is often expressed through various identities. One such identity involves the half-angle functions and helps us to simplify expressions involving sine and cosine. This article will demonstrate the proof that:

Step-by-Step Proof

Step 1: Use Half-Angle Identities

The half-angle identities are as follows:

[cosleft(frac{x}{2}right) sqrt{frac{1 cos x}{2}}]

[sinleft(frac{x}{2}right) sqrt{frac{1-cos x}{2}}]

Step 2: Express 1 - sinx

We can express (sin x) using the double-angle identity:

[sin x 2 sinleft(frac{x}{2}right) cosleft(frac{x}{2}right)]

Thus, we have:

[1 - sin x 1 - 2 sinleft(frac{x}{2}right) cosleft(frac{x}{2}right)]

Step 3: Rewrite in terms of half-angle functions

Using the Pythagorean identity, we know:

[cos^2left(frac{x}{2}right) - sin^2left(frac{x}{2}right) 1]

Therefore, we can express (1 - sin x) as:

[1 - sin x left[cosleft(frac{x}{2}right) - sinleft(frac{x}{2}right)right]^2]

Step 4: Square the right-hand side

Now, let's square (cosleft(frac{x}{2}right) - sinleft(frac{x}{2}right)):

[left[cosleft(frac{x}{2}right) - sinleft(frac{x}{2}right)right]^2 cos^2left(frac{x}{2}right) - 2 cosleft(frac{x}{2}right) sinleft(frac{x}{2}right) sin^2left(frac{x}{2}right)]

Using the Pythagorean identity again, we get:

[cos^2left(frac{x}{2}right) sin^2left(frac{x}{2}right) 1]

So we can rewrite the expression as:

[left[cosleft(frac{x}{2}right) - sinleft(frac{x}{2}right)right]^2 1 - 2 cosleft(frac{x}{2}right) sinleft(frac{x}{2}right))

But we know from Step 2 that:

[1 - 2 cosleft(frac{x}{2}right) sinleft(frac{x}{2}right) 1 - sin x)

Hence:

[left[cosleft(frac{x}{2}right) - sinleft(frac{x}{2}right)right]^2 1 - sin x)

Step 5: Relate to √(1 - sinx)

Therefore:

[sqrt{1 - sin x} left[cosleft(frac{x}{2}right) - sinleft(frac{x}{2}right)right])

Final Consideration:

The equality holds in the appropriate domain of (x) where both sides are non-negative. Thus, we can conclude:

[sqrt{1 - sin x} cosleft(frac{x}{2}right) - sinleft(frac{x}{2}right))

Conclusion:

This identity is useful in various mathematical and engineering problems, particularly in simplifying trigonometric expressions. The intervals must be considered to ensure the identity holds true.