Prove that: ( + )^2+( - )^2=4 ^2 x+y 2 - Vidyadip

Question

Prove that: $(\cos\text{x}+\cos\text{y})^2+(\sin\text{x}-\sin\text{y})^2=4\cos^2\frac{\text{x}+\text{y}}{2}$

✓

Answer

$\text{L.H.S.}=(\cos\text{x}+\cos\text{y})^2+(\sin\text{x}-\sin\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}$
$=(\cos^2\text{x}+\sin^2\text{x})+(\cos^2\text{y}+\sin^2\text{y})+2(\cos\text{x}\cos\text{y}-\sin\text{x}\sin\text{y})$
$=1+1+2\cos(\text{x}+\text{y})\ [\cos(\text{A}+\text{B})=(\cos\text{A}\cos\text{B}-\sin\text{A}\sin\text{B})]$
$=2+2\cos(\text{x}+\text{y})$
$=2[1+\cos(\text{x}+\text{y})]$
$=2\Big[1+2\cos^2\Big(\frac{\text{x}+\text{y}}{2}\Big)-1\Big]\ [\cos2\text{A}=2\cos^2\text{A}-1]$
$=4\cos^2\frac{\text{x}+\text{y}}{2}=\text{R.H.S.}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "(Each question 4 marks)" from TRIGONOMETRIC FUNCTIONS→TRIGONOMETRIC FUNCTIONS — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Find the equations of the hyperbola satisfying the given conditions. Foci $(\pm4,0),$ the latus rectum is of length $12$.

For any two sets A and B, prove that
$\text{A}-(\text{A}\cap\text{B})=\text{A} - \text{B}$

If $\sin(\alpha+\beta)=1$ and $\sin(\alpha-\beta)=\frac{1}{2}$ where $0\leq\alpha,\beta\leq\frac{\pi}{2}$ than find the values of $\tan(\alpha+2\beta)$and $\tan(2\alpha+\beta).$

Prove that: $\cos^32\text{x}+3\cos2\text{x}=4(\cos^6\text{x}-\sin^6\text{x})$

Find sum of these numbers in G.P. is $21$ and the sum of their aquares is $189$. Find the numbers.

Prove that the area of the parallelogram formed by the lines $a_1x + b_1y + c_1 = 0, a_1x + b_1y+ d_1 = 0, a_2x + b_2y + c_2 = 0, a_2x + b_2y + d_2 = 0$ is $\Big|\frac{(\text{d}_1-\text{c}_1)(\text{d}_2-\text{c}_2)}{\text{a}_1\text{b}_2-\text{a}_2\text{b}_1}\Big|$ sq.units. Deduce the condition for these lines to form a rhombus.

Find the mean and variance for the following frequency distribution

Classes	0-10	10-20	20-30	30-40	40-50
Frequencies	5	8	15	16	6

Prove that: $\frac{\cos4\text{A}+\cos3\text{A}+\cos2\text{A}}{\sin4\text{A}+\sin3\text{A}+\sin2\text{A}}=\cot3\text{A}$

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow1}\frac{\sqrt{5\text{x}-4}-\sqrt{\text{x}}}{\text{x}-1}$

Find the equation of the circle with radius $5$ whose centre lies on x-axis and passes through the point $(2, 3)$.