Question

find the value of (secA×cotA)^2 - (cosecA×cosA)^2
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Answer 1

$\huge\bold{\color{aqua}{Solution :-}}$

$\quad\bold{(sec\:A\:×\:cot\:A)²\:-\:(cosec\:A\:×\:cos\:A)² }$

$=\tiny\bold{\big(\frac{1}{cos\:A}\:×\:\frac{cos\:A}{sin\:A}\big)²\:-\:\big(\frac{1}{sin\:A}\:×\:cos\:A\big)² \qquad\tiny\therefore\orange{\begin{bmatrix}\diamond\:\: sec\:B=\frac{1}{cos\:B}\\\\\diamond\:\: cot\:B=\frac{cos\:B}{sin\:B}\\\\\diamond\:\: cosec\:B =\frac{1}{sin\:B}\end{bmatrix}}}$

$=\bold{(\frac{1}{\red{\cancel{\blue{cos\:A}}}}\:×\:\frac{\red{\cancel{\blue{cos\:A}}}}{sin\:A})²\:-\:(\frac{cos\:A}{sin\:A})²}$

$=\bold{(\frac{1}{sin\:A})²\:-\:(cot\:A)²\qquad\qquad\:\:\small\therefore\orange{\diamond\:\frac{cos\:B}{sin\:B}=cot\:B}}$

$=\bold{(cosec\:A)²\:-\:(cot\:A)²\qquad\quad\therefore\small\orange{\diamond\:\frac{1}{sin\:B}=cosec\:B}}$

$=\bold{cosec²A\:-\:cot²\:A}$

$=\bold{\green{\boxed{\pink{1}}}\qquad\qquad\therefore\small\orange{cosec²B-cot²B=1}}$