Math, asked by ravib225638, 6 months ago

if cosec titar+ cot titar=K then prove that cos titar=ke power 2 -1 bye ke power 2 +1​

Answers

Answered by VishnuPriya2801
11

Answer:-

Given:

Cosec θ + cot θ = k -- equation (1)

we know that,

cosec² θ - cot² θ = 1

  • - = (a + b)(a - b)

So,

⟶ (cosec θ + cot θ) (cosec θ - cot θ) = 1

⟶ k * (cosec θ - cot θ) = 1

[ From equation (1) ]

Cosec θ - cot θ = 1/k -- equation (2)

Add equations (1) and (2)

 \longrightarrow \sf \:  \csc \theta +  \cot \theta +  \csc \theta  - \cot \theta  = k +  \frac{1}{k}  \\ \\ \\\longrightarrow \sf \: 2  \csc \theta =  \frac{ {k}^{2} + 1 }{k}  \\ \\ \\  \longrightarrow \sf\csc \theta =  \frac{ {k}^{2} + 1 }{k} \times  \frac{1}{2}  \\ \\ \\ \longrightarrow  \boxed{\sf\csc \theta =  \frac{ {k}^{2} + 1 }{2k} }

  • cot θ = Cosec θ * cos θ

So substitute the value of cosec θ in equation (1).

⟶ Cosec θ + cosec θ * cos θ = k

⟶ cosec θ (1 + cos θ) = k

 \: \longrightarrow \sf \:  \frac{ {k}^{2} + 1 }{2k}  \bigg(1 +  \cos \theta \bigg) = k \\ \\ \\ \longrightarrow \sf \:1 +  \cos \theta =k \times   \frac{2k}{ {k}^{2}  + 1}  \\  \\\\ \longrightarrow \sf \: \cos \theta =  \frac{2 {k}^{2} }{ {k}^{2} + 1 }  - 1 \\  \\\\ \longrightarrow \sf \:\cos \theta =  \frac{2 {k}^{2}  - ( {k}^{2}  + 1)}{ {k}^{2}  + 1}  \\\\  \\ \longrightarrow \sf \:\cos \theta =  \frac{2 {k}^{2}  -  {k}^{2}   -  1}{ {k}^{2}  + 1} \\\\  \\ \longrightarrow  \boxed{\sf \:\cos \theta =  \frac{{k}^{2}   -  1}{ {k}^{2}  + 1}}

Hence Proved.

Answered by Anonymous
208

Step-by-step explanation:

Correct Question :

  • If cosec theta + cot theta = k prove that cos theta = k2-1 / k2 + 1

See the attachment

Pic is blur Sorry !!

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