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Question
(cot A − cot B)2 + (1 + cot A cot B)2 is equal to ______.
Options
sec2 A − cos2 A
sec2 A − cosec2 A
(1 + tan A)2 − (1 − cot A)2
cosec2 A − cosec2 В
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Solution
(cot A − cot B)2 + (1 + cot A cot B)2 is equal to cosec2 A − cosec2 В.
Explanation:
Solving,
⇒ (cot A − cot B)2 + (1 + cot A cot B)2
⇒ cot2 A + cot2 B − 2 cot A cot B + 1 + cot2 A . cot2 B + 2 cot A cot B
⇒ cot2 A + cot2 A . cot2 B + 1 + cot2 B
⇒ cot2 A(1 + cot2 B) + (1 + cot2 B)
⇒ (1 + cot2 B)(1 + cot2 A)
\[\Rightarrow \left(1 + \frac{\cos^2 B}{\sin^2 B}\right)\left(1 + \frac{\cos^2 A}{\sin^2 A}\right)\]
\[\Rightarrow \left(\frac{\sin^2 B + \cos^2 B}{\sin^2 B}\right)\left(\frac{\sin^2 A + \cos^2 A}{\sin^2 A}\right)\]
Substituting, sin2 θ + cos2 θ = 1, we get:
\[\Rightarrow \left(\frac{1}{\sin^2 B}\right)\left(\frac{1}{\sin^2 A}\right)\]
⇒ cosec2 B . cosec2 A
