If ( + )^2+( + )^2= ^2 ( - 2 ), write the value of . - Vidyadip

Question

If $(\cos\alpha+\cos\beta)^2+(\sin\alpha+\sin\beta)^2=\lambda\cos^2\Big(\frac{\alpha-\beta}{2}\Big),$ write the value of $\lambda.$

✓

Answer

We have, $\text{LHS}=(\cos\alpha+\cos\beta)^2+(\sin\alpha+\sin\beta)^2$ $=\ \Big[2\cos\Big(\frac{\alpha+\beta}{2}\Big)\cos\Big(\frac{\alpha-\beta}{2}\Big)\Big]^2+\Big[2\sin\Big(\frac{\alpha+\beta}{2}\Big)\cos\Big(\frac{\alpha-\beta}{2}\Big)\Big]^2$ $=\ 4\cos^2\Big(\frac{\alpha+\beta}{2}\Big)\cos^2\Big(\frac{\alpha-\beta}{2}\Big)+4\sin^2\Big(\frac{\alpha+\beta}{2}\Big)\cos^2\Big(\frac{\alpha-\beta}{2}\Big)$ $=\ 4\cos^2\Big(\frac{\alpha-\beta}{2}\Big)\Big[\cos^2\Big(\frac{\alpha+\beta}{2}\Big)+\sin^2\Big(\frac{\alpha+\beta}{2}\Big)\Big]$ $=\ 4\cos^2\Big(\frac{\alpha-\beta}{2}\Big)$ $[\because\ \cos^2\theta+\sin^2\theta=1]$ $\Rightarrow\ (\cos\alpha+\cos\beta)^2+(\sin\alpha+\sin\beta)^2=4\cos^2\Big(\frac{\alpha-\beta}{2}\Big)$ It is given that, $(\cos\alpha+\cos\beta)^2+(\sin\alpha+\cos\beta)^2=\lambda\cos^2\Big(\frac{\alpha-\beta}{2}\Big)$ On comparing, we get $\lambda=4$ $\therefore\ \lambda=4$

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 1 marks)" from Transformation Formulae→Transformation Formulae — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

A die is thrown, find the probability of getting: 2 or 4.

If A and B are two sets such that $\text{A}\subset\text{B},$ then find:

$\text{A}\cap\text{B}$

If A = {3, 6, 9, 12, 15, 18, 21}, B = {4, 8, 12,16, 20}, C = {2, 4, 6, 8, 10, 12, 14, 16}, D = {5, 10, 15, 20}, find:A - D

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow0}\frac{\sqrt{\text{x}+8}}{\sqrt{\text{x}}}$

P(A)	P(B)	P(A$\cap$B)	P(A$\cup$B)
0.35	_____	0.25	0.6

Wrie the coordinates of the foci of the hyperbola $9\text{x}^{2}-16\text{y}^{2}=144$

A group consist of 4 girls and 7 boys. In how many ways, a team of 5 members be selected, if the team has at least one boy and one girl ?

Check the validity of the following statements: q: 125 is a multiple of 5 and 7.

Determine the contrapositive of the following statements: Only if Max studies will he pass the test.

Evaluate the following limits in Exercise:$\lim\limits_{\text{x}\rightarrow0}\frac{\text{ax}+\text{b}}{\text{cx}+1}$