Question

sin square 10 + sin square 80 / sec square 20 - cot square 70​

Answer 1

Given Question :-

$\tt \:  \longrightarrow \: Evaluate \: \dfrac{ {sin}^{2}10 + {sin}^{2} 80 }{ {sec}^{2}20 - {cot}^{2} 70}$

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$\huge\bf \: \red{AηsωeR }$

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$\begin{gathered}\Large{\bold{\pink{\underline{Formula \: Used \::}}}} \end{gathered}$

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$(1). \: \boxed{ \red{ \bf \: sin(90 - x) = cosx}}$

$(2). \: \boxed{ \red{ \bf cot(90 - x) = tanx\bf \: }}$

$(3). \: \boxed{ \red{ \bf \: {sin}^{2} x + {cos}^{2}x = 1 }}$

$(4). \: \boxed{ \red{ \bf \: {sec}^{2} x - {tan}^{2}x = 1 }}$

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$\large\underline\purple{\bold{Solution :- }}$

$\tt \:  \longrightarrow \: \: \dfrac{ {sin}^{2}10 + {sin}^{2} 80 }{ {sec}^{2}20 - {cot}^{2} 70}$

$\tt \:  \longrightarrow \: \: \dfrac{ {sin}^{2}10 + {sin}^{2} (90 - 10) }{ {sec}^{2}20 - {cot}^{2} (90 - 20)}$

$\tt \:  \longrightarrow \: \: \dfrac{ {sin}^{2}10 + {cos}^{2} 10 }{ {sec}^{2}20 - {tan}^{2} 20}$

$\tt \:  \longrightarrow \: 1$

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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

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