If the vectors ( ^2A ) i + j + k , i + ( ^2B )+ k , i + j + ( ^2C ) k are coplanar, then find the value of cosec^2A+cosec^2B+cosec^2C.

Let: a = ( ^2A ) i + j , b = i +( ^2B) j + k and c = i + j +( ^2C) k We know that three vectors are coplanar if their scaler triple product is zero i.e., [ a b c ]=0 Here, [ a b c ]=0 ^2A&1&1 1& ^2B&1 1&1& ^2C =0 ^2A [ ( ^2B ^2C ) -1 ]-1 ( ^2C-1 )+1 (1- ^2B )=0 ^2A ^2B ^2C- ^2A- ^2C+1+1- ^2B=0 (1+ ^2A ) (1+ ^2B ) (1+ ^2C ) - (1+ ^2A )- (1+ ^2C )+1=1- (1+ ^2B )=0 1+ ^2A+ ^2B+ ^2C+ ^2A ^2B + ^B ^2C+ ^2C ^2A+ ^2A ^2B ^C1 - ^2A-1- ^2C ^2A ^2B+ ^2B ^2C+ ^2C ^2A + ^2A ^2B ^2C=0 ^2A ^2B+ ^2B ^2C+ ^2C ^2A =- ^2A ^2B ^2C ^2A ^2B ^2C+ ^2C ^2A ^2A ^2B ^2C =-1 ^2C+ ^2A+ ^2B=-1 ^2C-1+cosec^2A-1+cosec^2B-1=-1 ^2A+cosec^2B+cosec^2C=2

If the vectors ( ^2A ) i + j + k , i + ( ^2B )+ k , i + j + ( ^2C…

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Question

If the vectors $\big(\sec^2\text{A}\big)\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}},\hat{\text{i}}+\big(\sec^2\text{B}\big)+\hat{\text{k}},\hat{\text{i}}+\hat{\text{j}}+\big(\sec^2\text{C}\big)\hat{\text{k}}$ are coplanar, then find the value of $\text{cosec}^2\text{A}+\text{cosec}^2\text{B}+\text{cosec}^2\text{C}.$

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Answer

Let: $\vec{\text{a}}=\big(\sec^2\text{A}\big)\hat{\text{i}}+\hat{\text{j}},\vec{\text{b}}=\hat{\text{i}}+(\sec^2\text{B})\hat{\text{j}}+\hat{\text{k}}$ and $\vec{\text{c}}=\hat{\text{i}}+\hat{\text{j}}+(\sec^2\text{C})\hat{\text{k}}$
We know that three vectors are coplanar if their scaler triple product is zero i.e., $\big[\vec{\text{a}}\vec{\text{b}}\vec{\text{c}}\big]=0$
Here, $\big[\vec{\text{a}}\vec{\text{b}}\vec{\text{c}}\big]=0$
$\begin{vmatrix}\sec^2\text{A}&1&1\\1&\sec^2\text{B}&1\\1&1&\sec^2\text{C} \end{vmatrix}=0$
$\Rightarrow\sec^2\text{A}\big[\big(\sec^2\text{B}\times\sec^2\text{C}\big)\\-1\big]-1\big(\sec^2\text{C}-1\big)+1\big(1-\sec^2\text{B}\big)=0$
$\Rightarrow\sec^2\text{A}\sec^2\text{B}\sec^2\text{C}-\sec^2\text{A}-\sec^2\text{C}+1+1-\sec^2\text{B}=0$
$\Rightarrow\big(1+\tan^2\text{A}\big)\big(1+\tan^2\text{B}\big)\big(1+\tan^2\text{C}\big)\\-\big(1+\tan^2\text{A}\big)-\big(1+\tan^2\text{C}\big)+1=1-\big(1+\tan^2\text{B}\big)=0$
$\Rightarrow1+\tan^2\text{A}+\tan^2\text{B}+\tan^2\text{C}+\tan^2\text{A}\tan^2\text{B}\\+\tan^\text{B}\tan^2\text{C}+\tan^2\text{C}\tan^2\text{A}+\tan^2\text{A}\tan^2\text{B}\tan^\text{C}1\\-\tan^2\text{A}-1-\tan^2\text{C}$
$\tan^2\text{A}\tan^2\text{B}+\tan^2\text{B}\tan^2\text{C}+\tan^2\text{C}\tan^2\text{A}\\+\tan^2\text{A}\tan^2\text{B}\tan^2\text{C}=0$
$\Rightarrow\tan^2\text{A}\tan^2\text{B}+\tan^2\text{B}\tan^2\text{C}+\tan^2\text{C}\tan^2\text{A}\\=-\tan^2\text{A}\tan^2\text{B}\tan^2\text{C}$
$\Rightarrow\frac{\tan^2\text{A}\tan^2\text{B}\tan^2\text{C}+\tan^2\text{C}\tan^2\text{A}}{\tan^2\text{A}\tan^2\text{B}\tan^2\text{C}}=-1$
$\Rightarrow\cot^2\text{C}+\cot^2\text{A}+\cot^2\text{B}=-1$
$\Rightarrow\text{cosec}^2\text{C}-1+\text{cosec}^2\text{A}-1+\text{cosec}^2\text{B}-1=-1$
$\therefore\text{cosec}^2\text{A}+\text{cosec}^2\text{B}+\text{cosec}^2\text{C}=2$

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