cos(x + 3π/2) + cos(x - 3π/2) = (cos x)(cos ) - (sin x)(sin 3π/2) + (cos x)(cos 3π/2) + (sin x)(sin ) = (cos…

Explanation:

Step1: Apply cosine addition and subtraction formulas

The cosine addition formula is $\cos(A + B)=\cos A\cos B-\sin A\sin B$ and the cosine - subtraction formula is $\cos(A - B)=\cos A\cos B+\sin A\sin B$. Here $A = x$ and $B=\frac{3\pi}{2}$. So $\cos(x+\frac{3\pi}{2})+\cos(x - \frac{3\pi}{2})=[\cos x\cos\frac{3\pi}{2}-\sin x\sin\frac{3\pi}{2}]+[\cos x\cos\frac{3\pi}{2}+\sin x\sin\frac{3\pi}{2}]$.

Step2: Evaluate trigonometric values

We know that $\cos\frac{3\pi}{2}=0$ and $\sin\frac{3\pi}{2}=- 1$. Substituting these values, we get $[\cos x\times0-\sin x\times(-1)]+[\cos x\times0+\sin x\times(-1)]$.

Step3: Simplify the expression

$[\cos x\times0-\sin x\times(-1)]+[\cos x\times0+\sin x\times(-1)]=\sin x-\sin x$.

Step4: Final simplification

$\sin x-\sin x = 0$.

Answer:

The first blank in the first - line is $\frac{3\pi}{2}$, the second blank in the first - line is $\frac{3\pi}{2}$, the first blank in the second - line is $0$, the second blank in the second - line is $-1$, the blank in the third - line is $\sin x$.