It is given that cos ( θ − u ) − a , cos ( θ − β ) − b What is sin 2 ( α − β ) + 2 ab cos ( α − β ) equal to ?

Question

It is given that cos ( θ − u ) − a , cos ( θ − β ) − b What is sin 2 ( α − β ) + 2 ab cos ( α − β ) equal to ?, a 2 + b 2, a 2 − b 2, b 2 − a 2, − ( a 2 + b 2 )

MARKS App · Accepted Answer

$\begin{aligned} & \sin ^{2}(\alpha-\beta)+2 a b \cos (\alpha-\beta)=1-\cos ^{2}(\alpha-\beta)+2 a b \cos \ &(\alpha-\beta)=1-\cos (\alpha-\beta)[\cos (\alpha-\beta)-2 a b] \ &=1-\left(a b+\sqrt{1-a^{2}} \sqrt{1-b^{2}}ight) \ & \quad\left[a b+\sqrt{1-a^{2}} \sqrt{1-b^{2}}-2 a bight] \ &=1-\left(\sqrt{1-a^{2}} \sqrt{1-b^{2}}+a bight)\left(\sqrt{1-a^{2}} \sqrt{1-b^{2}}-a bight) \ &=1-\left[\left(\sqrt{1-a^{2}} \sqrt{1-b^{2}}ight)^{2}-(a b)^{2}ight] \ &=1-\left[\left(1-a^{2}ight)\left(1-b^{2}ight)-a^{2} b^{2}ight) \ &=1-\left(1-b^{2}-a^{2}+a^{2} b^{2}-a^{2} b^{2}ight) \ &=1-1+b^{2}+a^{2}=a^{2}+b^{2}

Search any question & find its solution

Looking for more such questions to practice?