If alpha + beta - gamma = pi , then {sin ^2}a | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Given $\alpha + \beta - \gamma = \pi ,$ so $\gamma = \alpha + \beta - \pi .$Consider the expression $E = \sin^2 \alpha + \sin^2 \beta - \sin^2 \ga

Vedclass · Accepted Answer

$2\sin \alpha \sin \beta \cos \gamma $

Vedclass · Answer

(A) Given $\alpha + \beta - \gamma = \pi ,$ so $\gamma = \alpha + \beta - \pi .$Consider the expression $E = \sin^2 \alpha + \sin^2 \beta - \sin^2 \gamma .$Using the identity $\sin^2 A - \sin^2 B = \sin(A - B)\sin(A + B),$$E = \sin^2 \alpha + \sin(\beta - \gamma)\sin(\beta + \gamma).$Since $\beta - \gamma = \pi - \alpha,$ we have $\sin(\beta - \gamma) = \sin(\pi - \alpha) = \sin \alpha.$Also,$\beta + \gamma = \beta + (\alpha + \beta - \pi) = \alpha + 2\beta - \pi.$Alternatively,using $\gamma = \alpha + \beta - \pi,$$E = \sin^2 \alpha + \sin^2 \beta - \sin^2(\alpha + \beta - \pi) = \sin^2 \alpha + \sin^2 \beta - \sin^2(\alpha + \beta).$Using $\sin^2 A - \sin^2 B = \sin(A-B)\sin(A+B),$$E = \sin^2 \alpha - \sin(\alpha + \beta - \beta)\sin(\alpha + \beta + \beta) = \sin^2 \alpha - \sin \alpha \sin(\alpha + 2\beta).$$E = \sin \alpha [\sin \alpha - \sin(\alpha + 2\beta)].$Using $\sin C - \sin D = 2\cos(\frac{C+D}{2})\sin(\frac{C-D}{2}),$$E = \sin \alpha [2\cos(\alpha + \beta)\sin(-\beta)] = -2\sin \alpha \sin \beta \cos(\alpha + \beta).$Since $\alpha + \beta = \pi + \gamma,$ $\cos(\alpha + \beta) = \cos(\pi + \gamma) = -\cos \gamma.$Therefore,$E = -2\sin \alpha \sin \beta (-\cos \gamma) = 2\sin \alpha \sin \beta \cos \gamma.$

If $\alpha + \beta - \gamma = \pi ,$ then ${\sin ^2}\alpha + {\sin ^2}\beta - {\sin ^2}\gamma = $

Similar Questions

The number of integral values of $k$,for which the equation $7 \cos x + 5 \sin x = 2k + 1$ has a solution,is

The minimum value of $8 \cos^2 x + 18 \sec^2 x$ for all $x \in R$ wherever it is defined,is:

The minimum value of $9 \tan^2 \theta + 4 \cot^2 \theta$ is

If $10 \sin^4 \alpha + 15 \cos^4 \alpha = 6$,then $16 \tan^6 \alpha + 27 \cot^6 \alpha =$

If $A+B+C=4S$,then $\cos (2S-A)+\cos (2S-B)-\cos (2S-C)-\cos 2S=$

Vedclass Test Series

Exam Paper Generator

Online Exam Module