Question

cosec²θ / cosec θ -1 - cosec²θ / cosecθ + 1 = 2 sec²θ
prove it plsssssssssssssss​

Answer 1

Question :-

$\bf \:Prove \: that \: \dfrac{ {cosec}^{2}θ }{cosecθ - 1} - \dfrac{ {cosec}^{2}θ }{cosecθ + 1} = {2sec}^{2} θ$

Solution :-

$\bf \:Consider \: \dfrac{ {cosec}^{2}θ }{cosecθ - 1} - \dfrac{ {cosec}^{2}θ }{cosecθ + 1}$

$\bf\implies \: {cosec}^{2} θ(\dfrac{ 1 }{cosecθ - 1} - \dfrac{ 1 }{cosecθ + 1})$

$\bf\implies \: {cosec}^{2} θ(\dfrac{cosecθ + 1 - cosecθ + 1}{(cosecθ + 1)(cosecθ - 1)} )$

$\bf\implies \:\dfrac{ {2cosec}^{2} θ}{ {cosec}^{2}θ - 1 }$

$\bf\implies \:\dfrac{2 {cosec}^{2}θ }{ {cot}^{2}θ }$

$\bf\implies \:2 \times \dfrac{1}{ {sin}^{2}θ } \times \dfrac{ {sin}^{2} θ}{ {cos}^{2}θ }$

$\bf\implies \:\dfrac{2}{ {cos}^{2} θ}$

$\bf\implies \: {2sec}^{2} θ$

$\large{\boxed{\boxed{\bf{Hence, Proved}}}}$

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❥︎Additional Information:-

Relationship between sides and T ratios

sin θ = Opposite Side/Hypotenuse

cos θ = Adjacent Side/Hypotenuse

tan θ = Opposite Side/Adjacent Side

sec θ = Hypotenuse/Adjacent Side

cosec θ = Hypotenuse/Opposite Side

cot θ = Adjacent Side/Opposite Side

❥︎Reciprocal Identities

cosec θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

sin θ = 1/cosec θ

cos θ = 1/sec θ

tan θ = 1/cot θ

❥︎Co-function Identities

sin (90°−x) = cos x

cos (90°−x) = sin x

tan (90°−x) = cot x

cot (90°−x) = tan x

sec (90°−x) = cosec x

cosec (90°−x) = sec x

❥︎Fundamental Trigonometric Identities

sin²θ + cos²θ = 1

sec²θ - tan²θ = 1

cosec²θ - cot²θ = 1

$</p><p>\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\bf Trigonometry\: Table \\ \begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm \infty \\ \\ \rm cosec A & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \rm cot A & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0\end{array}}}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$