Question

If cosec 0 + cot 0 = k then find the value of cos in terms of k. plz answer this question with step by step
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Answer 1

Given :

cosecθ + cotθ = k

To find :

The value of cos in terms of k.

Step-by-step explanation:

• We know that,

       $cosecx=\frac{1}{sinx}$ , $cotx=\frac{cosx}{sinx}$

• By substituting this in the given equation,

      $\frac{1}{sinθ}+\frac{cosθ}{sinθ} =k$

• By taking the LCM,

       $\frac{1+cosθ}{sinθ}=k$

• By cross multiplying,

       $1+cos$θ = $ksin$θ

• By squaring on both sides,

       $(1+cos$ θ$)^{2}$ = $k^{2} sin^{2}$θ

• We know that,

       $sin^{2}$θ = $1-cos^{2}$θ

• $cos^{2}$θ can be written as

      $(1+cos$θ$)(1-cos$θ$)$

• By substituting these values in the above equation,

      $(1+cos$θ$)^{2}$ = $k^{2}$$(1+cos$θ$)(1-cos$θ$)$

• By cancelling out the common terms,

      $(1+cos$θ$)$ = $k^{2}$$(1-cos$θ$)$

• Take $k^{2}$ inside,

       $(1+cos$θ$)$=$k^{2} -k^{2} cos$θ

• By taking $cos^{2}$θ to the other side,

       $1+cos$θ$+k^{2} cos$θ=$k^{2}$

• Take 1 to the other side,

       $cos$θ$+k^{2} cos$θ=$k^{2} -1$

• Take cosθ as common,

       $cos$θ $(1+k^{2}$$)$ = $k^{2} -1$

• Keep cosθ on the same side,

       $cos$θ=$\frac{k^{2}-1 }{k^{2}+1 }$

Final answer :

The value of cosθ is $\frac{k^{2}-1 }{k^{2}+1 }$ .