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Show that ^2 , ( , + , A) , + , ^2 ,( , + , B) , - , 2 , ,(A

Question

Show that
$sin^{2} (θ + A) + sin^{2} (θ + B) - 2 cos (A - B) sin (θ + A) sin (θ + B)$
is independent of $θ or = sin^{2} (A - B)$

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Elijah Discussed

Show that

sin^{2} (θ + A) + sin^{2} (θ + B) - 2 cos (A - B) sin (θ + A) sin (θ + B)

is independent of

θ or = sin^{2} (A - B)

10 mins ago

Discuss this question LIVE

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Text solutionVerified

Let

θ

+ A = x,

θ

+ B = y

∴

A - B = x - y

....(1)

L.H.S. =

sin^{2} x + sin^{2} y - 2 sin x sin y cos (x - y)

= sin^{2} x + sin^{2} y - 2 sin x sin y (cos x cos y + sin x sin y)

= sin^{2} x (1 - sin^{2} y) + sin^{2} y (1 - sin^{2} x) - 2 sin x sin y cos x cos y

= sin^{2} x co s^{2} y + co s^{2} x sin^{2} y - 2 sin x sin y cos x cos y

= (sin x cos y - cos x sin y)^{2} = sin^{2} (x - y)

= sin^{2} (A - B)

, by (1)

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Question Text	Show that $sin^{2} (θ + A) + sin^{2} (θ + B) - 2 cos (A - B) sin (θ + A) sin (θ + B)$ is independent of $θ or = sin^{2} (A - B)$
Answer Type	Text solution:1
Upvotes	150