Question

If a cot A + b cosec A=p and b cot A +a cosec A =q then p2-q2 is equal to :
(A) a2-b2 (B) b2-a2 (C) a2+b2 (D) b-a
[A=theta]

Answer 1

option B is correct

Answer 2

The required option "B. $b^{2}-a^{2}$" is correct.

Step-by-step explanation:

We have,

$a\cot \theta+b\csc \theta=p$                 .....(1)

and $b\cot \theta+a\csc \theta=q$        .....(2)

To find, the value of $p^{2} -q^{2} =?$

Squaring (1) and (2) and subtracting them, we get

$(a^{2} \cot ^{2} \theta+b^{2} \csc^{2} \theta+2ab\cos \theta\csc \theta) -(b^{2} \cot ^{2} \theta+a^{2}\csc^{2}  \theta+2ab\cos \theta\csc \theta)=p^{2} -q^{2}$

⇒ $(a^{2} \cot ^{2} \theta+b^{2} \csc^{2} \theta) -(b^{2} \cot ^{2} \theta+a^{2} \csc^{2}  \theta)=p^{2} -q^{2}$

⇒$a^{2} (\cot ^{2} \theta-\csc^{2}\theta )+ b^{2} (\csc ^{2} \theta-\cot^{2}\theta) =p^{2} -q^{2}$

⇒ $a^{2} (-1)+ b^{2} (1) =p^{2} -q^{2}$

[∵ $\csc ^{2} \theta-\cot^{2}\theta=1$]

⇒ $b^{2}-a^{2}=p^{2} -q^{2}$

Hence, the required option "B. $b^{2}-a^{2}$" is correct.