(A) : If A = 350, B = 150, C = 400, then tanA tanB + tanB tanC + tanC tanA = 1(R) : If A + B + C =900, then ∑tan A tan B=1Then which one of the following is true ?

A=35o, B=15o, C=40o A+B+C=90∘⟹A+B=90∘−C tan⁡(A+B)=tan⁡90∘−C ⟹tan⁡(A+B)=cot⁡C ⟹tan⁡A+tan⁡B1−tan⁡Atan⁡B=1tan⁡C ⟹tan⁡Atan⁡C+tan⁡Btan⁡C=1−tan⁡Atan⁡B ⟹tan⁡Atan⁡B+tan⁡Btan⁡C+tan⁡Ctan⁡A=1 ∴ ∑tan⁡Atan⁡B=1

(A) : If A = 350, B = 150, C = 400, then tanA tanB + tanB tanC + tanC tanA = 1(R) : If A + B + C =900, then ∑tan A tan B=1Then which one of the following is true ?

A=35o, B=15o, C=40o A+B+C=90∘⟹A+B=90∘−C tan⁡(A+B)=tan⁡90∘−C ⟹tan⁡(A+B)=cot⁡C ⟹tan⁡A+tan⁡B1−tan⁡Atan⁡B=1tan⁡C ⟹tan⁡Atan⁡C+tan⁡Btan⁡C=1−tan⁡Atan⁡B ⟹tan⁡Atan⁡B+tan⁡Btan⁡C+tan⁡Ctan⁡A=1 ∴ ∑tan⁡Atan⁡B=1

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$(A) :$ If $A = 35^{0}, B = 15^{0}, C = 40^{0},$ then $tanA tanB + tanB tanC + tanC tanA = 1$
$(R) :$ If $A + B + C = 90^{0,}$ then $\sum$ $\tan A \tan B = 1$
Then which one of the following is true ?

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$A$ is true, $R$ is false

$A$ is false , $R$ is true

$A$ is true, $R$ is true but $R$ is not correct explanation of $A$

$A$ is true, $R$ is true and $R$ is correct explanation of $A$

answer is A.

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Detailed Solution

$\begin{array}{l} A = 35^{o}, B = 15^{o}, C = 40^{o} A + B + C = 90^{\circ} \\ ⟹ A + B = 90^{\circ} - C \tan (A + B) = \tan (90^{\circ} - C) ⟹ \tan (A + B) = \cot C ⟹ \frac{\tan A + \tan B}{1 - \tan A \tan B} = \frac{1}{\tan C} ⟹ \tan A \tan C + \tan B \tan C = 1 - \tan A \tan B ⟹ \tan A \tan B + \tan B \tan C + \tan C \tan A = 1 ∴ \sum \tan A \tan B = 1 \end{array}$