Question

(cosec theta -sin theta )(sec theta-cos theta)=1/tan theta+ cot theta​

Answer 1

Identities Used -

$\longmapsto \: \begin{gathered}\:{\underline{\boxed{\bf{\blue{{\tt \: \: cot\theta \: = \: \dfrac{cos \: \theta}{sin \: \theta} }}}}}} \\ \end{gathered}$

$\longmapsto \: \begin{gathered}\:{\underline{\boxed{\bf{\blue{{\tt \: \: tan\theta \: = \: \dfrac{cos \: \theta}{sin \: \theta} }}}}}} \\ \end{gathered}$

$\longmapsto \: \begin{gathered}\:{\underline{\boxed{\bf{\blue{{\tt \: \: sec\theta \: = \: \dfrac{1}{cos \: \theta} }}}}}} \\ \end{gathered}$

$\longmapsto \: \begin{gathered}\:{\underline{\boxed{\bf{\blue{{\tt \: \: cosec\theta \: = \: \dfrac{1}{sin \: \theta} }}}}}} \\ \end{gathered}$

$\longmapsto \: \begin{gathered}\:{\underline{\boxed{\bf{\blue{{\tt {sin}^{2} \theta + {cos}^{2}\theta = 1 }}}}}} \\ \end{gathered}$

Prove that -

$\bf \: (sec\theta - cos\theta)(cosec\theta - sin\theta) = \dfrac{1}{tan\theta + cot\theta}$

Solution

Solution Consider

$\longmapsto \: \bf \: LHS$

$\bf \: (sec\theta - cos\theta)(cosec\theta - sin\theta)$

$\rm :\implies\:(\dfrac{1}{cos\theta} -cos\theta )(\dfrac{1}{sin\theta} - sin\theta)$

$\rm :\implies\:\dfrac{1 - {cos}^{2}\theta }{cos\theta} \times \dfrac{1 - {sin}^{2}\theta }{sin\theta}$

$\rm :\implies\:\dfrac{ {cos}^{2} \theta}{sin\theta} \times \dfrac{ {sin}^{2}\theta }{cos\theta}$

$\rm :\implies\:sin\theta \: cos\theta$

Consider

$\longmapsto \: \bf \: RHS$

$\bf \: \dfrac{1}{tan\theta \: + \: cot\theta}$

$\rm :\implies\:\dfrac{1}{\dfrac{sin\theta}{cos\theta} + \dfrac{cos\theta}{sin\theta} }$

$\rm :\implies\:\dfrac{1}{\dfrac{ {sin}^{2}\theta + {cos}^{2}\theta }{sin\theta \: cos\theta} }$

$\rm :\implies\:sin\theta \: cos\theta$

$\bf\implies \:LHS \: = \: RHS$

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

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