Question
Prove that:
$(\cos\text{x}+\cos\text{y})^2+(\sin\text{x}-\sin\text{y})^2=4\cos^2\frac{\text{x}+\text{y}}{2}$
$(\cos\text{x}+\cos\text{y})^2+(\sin\text{x}-\sin\text{y})^2=4\cos^2\frac{\text{x}+\text{y}}{2}$
$=\cos^2\text{x}+\cos^2\text{y}+2\cos\text{x}\cos\text{y}+\sin^2\text{x}+\sin^2\text{y}-2\sin\text{x}\sin\text{y}$
$=(\cos^2\text{x}+\sin^2\text{x})+(\cos^2\text{y}+\sin^2\text{y})+2(\cos\text{x}\cos\text{y}-\sin\text{x}\sin\text{y})$
$=1+1+2\cos(\text{x}+\text{y})\ [\cos(\text{A}+\text{B})=(\cos\text{A}\cos\text{B}-\sin\text{A}\sin\text{B})]$
$=2+2\cos(\text{x}+\text{y})$
$=2[1+\cos(\text{x}+\text{y})]$
$=2\Big[1+2\cos^2\Big(\frac{\text{x}+\text{y}}{2}\Big)-1\Big]\ [\cos2\text{A}=2\cos^2\text{A}-1]$
$=4\cos^2\frac{\text{x}+\text{y}}{2}=\text{R.H.S.}$
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$(\text{x}+\text{y})+(\text{x}^2+\text{xy}+\text{y}^2)+(\text{x}^3+\text{x}^2\text{y}+\text{xy}^2+\text{y})+\ ...\text{ to n terms;}$
| Column A | Column B | ||
| a. | The polar form of $\text{i}+\sqrt{3}$ is | i. | Perpendicular bisector of segment joining (– 2, 0) and (2, 0). |
| b. | The amplitude of $-1+\sqrt{-3}$ is | ii. | On or outside the circle having centre at (0, – 4) and radius 3. |
| c. | If |z + 2| = |z - 2|, then locus of z is | iii. | $\frac{2\pi}{3}$ |
| d. | If |z + 2i| = |z - 2i|, then locus of z is | iv. | Perpendicular bisector of segment joining (0, – 2) and (0, 2). |
| e. | Region represented by $|\text{z}+4\text{i}|\geq3$ is | v. | $2\Big(\cos\frac{\pi}{6}+\text{i}\sin\frac{\pi}{6}\Big)$ |
| f. | Region represented by $|\text{z}+4|\leq3$ is | vi. | On or inside the circle having centre (– 4, 0) and radius 3 units. |
| g. | Conjugate of $\frac{1+2\text{i}}{1-\text{i}}$ lies in | vii. | First quadrant. |
| h. | Reciprocal of 1 - i lies in | viii. | Third quadrant. |
| (i) | $((\text{A}'\cup\text{B}')-\text{A})'$ | (a) | $\text{A} - \text{B}$ |
| (ii) | $[\text{B}'\cup(\text{B}'-\text{A})]'$ | (b) | $\text{A}$ |
| (iii) | $(\text{A} - \text{B}) - (\text{B} - \text{C})$ | (c) | $\text{B}$ |
| (iv) | $(\text{A}-\text{B})\cap(\text{C}-\text{B})$ | (d) | $(\text{A}\times\text{B})\cap(\text{A}\times\text{C})$ |
| (v) | $\text{A}\times(\text{B}\cap\text{C})$ | (e) | $(\text{A}\times\text{B})\cup(\text{A}\times\text{C})$ |
| (vi) | $\text{A}\times(\text{B}\cup\text{C})$ | (f) | $(\text{A}\cap\text{C})-\text{B}$ |