h) $$ \frac{1-\cos ^{2} heta}{\cos ^{2} heta+2 \cos heta+1}= $$

Question

h) $$ \frac{1-\cos ^{2} 	heta}{\cos ^{2} 	heta+2 \cos 	heta+1}= $$

UpStudy AI · Accepted Answer

We start with the expression:

(1 – cos²θ) / (cos²θ + 2cosθ + 1)

Step 1. Recognize that 1 – cos²θ is sin²θ, so the numerator becomes sin²θ.

Step 2. Notice that the denominator cos²θ + 2cosθ + 1 factors as (cosθ + 1)².

Thus, the expression simplifies to:

sin²θ / (cosθ + 1)²

Step 3. This can be rewritten as:

(sinθ / (cosθ + 1))²

Step 4. Using the half-angle identity:
  tan(θ/2) = sinθ / (1 + cosθ)

we see that:

(sinθ / (1 + cosθ))² = tan²(θ/2)

Therefore, the simplified form of the expression is:

tan²(θ/2)

So, the final answer is:

tan²(θ/2)
The expression simplifies to $$ 	an^2\left(\frac{	heta}{2}ight) $$.

h) \( \frac{1-\cos ^{2} \theta}{\cos ^{2} \theta+2 \cos \theta+1}= \)