Question

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(cos ∅ × cosec theta - sin ∅ × sec theta) / (cos ∅ + sin ∅) = cosec ∅ - sec ∅​


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Answer 1

Given :

• $\sf{LHS = \dfrac{cos \theta \: cosec \theta - sin\theta \: sec\theta}{cos \theta + sin \theta} }$

• $\sf{RHS = cosec \theta - sec \theta }$

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To Prove :

• L.H.S = R.H.S

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$\underline{\textbf{\textsf{ Proof :-}}}$

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• $\sf{LHS = \dfrac{cos \theta \: cosec \theta - sin\theta \: sec\theta}{cos \theta + sin \theta} }$

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$\sf{~~~~~:\implies \dfrac{cos \theta \dfrac{1}{sin \theta} - sin \theta \dfrac{1}{cos \theta} }{cos \theta + sin \theta} }$

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$\sf{~~~~~:\implies \dfrac{ \dfrac{cos \theta}{sin \theta} - \dfrac{sin \theta}{cos \theta} }{cos \theta + sin \theta} }$

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$\sf{~~~~~:\implies \dfrac{ \dfrac{ {cos}^{2} \theta - {sin}^{2} \theta }{sin \theta \: cos \theta} }{(cos \theta +sin \theta) } }$

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$\sf{~~~~~:\implies \dfrac{ \dfrac{ \cancel{(cos \theta + sin\theta})(cos\theta - sin\theta}{sin\theta \: cos\theta} }{ \cancel{(cos \theta + sin\theta)} }}$

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$\sf{~~~~~:\implies \dfrac{cos \theta - sin \theta}{sin \theta \: cos \theta} }$

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$\sf{~~~~~:\implies \dfrac{1}{sin \theta} - \dfrac{1}{cos \theta} }$

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$~~~~~:\implies{\pmb{\underline{\boxed{\pink{\frak{cosec \theta - sec \theta }}}}}}$★

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Hence,

• LSH = RHS

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${\rule{190pt}{2pt}}$

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$\underline{\textbf{\textsf{Remember that :-}}}$

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• $\sf{ cosec\theta = \dfrac{1}{sin\theta} }$

• $\sf{ sec\theta = \dfrac{1}{cos\theta} }$

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${\rule{190pt}{2pt}}$

Answer 2

Answer: (3) 2m

Solution:

Given,

sin θ + cos θ = m

sec θ + cosec θ = n

n(m + 1)(m – 1) = n(m2 – 1)

= (sec θ + cosec θ) [(sin θ + cos θ)2 – 1]

= (sec θ + cosec θ)[sin2θ + cos2θ + 2 sin θ cos θ – 1]

= (sec θ + cosec θ)[1 + 2 sin θ cos θ – 1]

= (sec θ + cosec θ)(2 sin θ cos θ)

= sec θ(2 sin θ cos θ) + cosec θ(2 sin θ cos θ)

= 2 sin θ + 2 cos θ

= 2(sin θ + cos θ)

= 2m