1. Given function

We have

$$y(\theta)=\frac{2\cos\theta+\cos2\theta}{\cos3\theta+4\cos2\theta+5\cos\theta+2}.$$

We need to find

$$y''\left(\frac{\pi}{2}\right)+y'\left(\frac{\pi}{2}\right)+y\left(\frac{\pi}{2}\right).$$

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2. Simplify the expression using trigonometric identities

Let $c=\cos\theta.$ Then $\cos2\theta=2c^2-1, \qquad \cos3\theta=4c^3-3c.$

So the numerator becomes

$$2\cos\theta+\cos2\theta=2c+(2c^2-1)=2c^2+2c-1.$$

The denominator becomes

$$\cos3\theta+4\cos2\theta+5\cos\theta+2 = (4c^3-3c)+4(2c^2-1)+5c+2.$$

Simplify:

$$=4c^3+8c^2+2c-2.$$

Factor it:

$$4c^3+8c^2+2c-2=2(2c^3+4c^2+c-1).$$

Now,

$$2c^3+4c^2+c-1=(c+1)(2c^2+2c-1),$$

so

$$4c^3+8c^2+2c-2=2(c+1)(2c^2+2c-1).$$

Hence

$$y=\frac{2c^2+2c-1}{2(c+1)(2c^2+2c-1)}=\frac{1}{2(c+1)}$$

whenever the cancellation is valid. Therefore,

$$y(\theta)=\frac{1}{2(1+\cos\theta)}.$$

Using $1+\cos\theta=2\cos^2\frac{\theta}{2},$ we get

$$y=\frac{1}{4\cos^2\frac{\theta}{2}}=\frac{1}{4}\sec^2\frac{\theta}{2}.$$

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3. Differentiate

We use $y=\frac{1}{4}\sec^2\frac{\theta}{2}.$

First derivative

Let $u=\frac{\theta}{2}.$ Then $y=\frac{1}{4}\sec^2 u.$

Now,

$$=2\sec^2 u\tan u\cdot \frac12 =\sec^2 u\tan u.$$

So

$$y'=\frac14\sec^2\frac{\theta}{2}\tan\frac{\theta}{2}.$$

Second derivative

Write

$$y'=\frac14\sec^2 u\tan u, \qquad u=\frac\theta2.$$

Then

$$y''=\frac14\cdot \frac12 \frac{d}{du}(\sec^2 u\tan u) =\frac18\left[(\sec^2 u)'\tan u+\sec^2 u(\tan u)'\right].$$

Now,

$$(\sec^2 u)'=2\sec^2 u\tan u, \qquad (\tan u)'=\sec^2 u.$$

Thus

$$y''=\frac18\left[2\sec^2 u\tan^2 u+\sec^4 u\right].$$

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4. Evaluate at $\theta=\frac{\pi}{2}$

Then $u=\frac{\theta}{2}=\frac{\pi}{4},$ so $\tan\frac{\pi}{4}=1, \qquad \sec^2\frac{\pi}{4}=2, \qquad \sec^4\frac{\pi}{4}=4.$

Therefore,

$$y\left(\frac{\pi}{2}\right)=\frac14\cdot 2=\frac12,$$ $$y'\left(\frac{\pi}{2}\right)=\frac14\cdot 2\cdot 1=\frac12,$$ $$y''\left(\frac{\pi}{2}\right)=\frac18\left[2\cdot 2\cdot 1^2+4\right] =\frac18(4+4)=1.$$

Hence,

$$y''+y'+y=1+\frac12+\frac12=2.$$

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5. Check options

The value is $2.$ So the correct option is:

D: $2$

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6. Comparison with stored answer

Stored correct answer: D

Our derived answer: D

So they agree.