Question

If sec x + tan x = k, then the value of sin
x is
​

Answer 1

Answer:Sinx=kCosx-1

Step-by-step explanation:

Secx+ Tanx =k

Or, (1/Cosx)+(Sinx/Cosx)=k

Or,(1+Sinx)/Cosx=k

Or, 1+Sinx=k Cosx

Or, Sinx=kCosx-1

Answer 2

Answer:

secx + tanx = k   → (1)

From identity sec²x-tan²x=1

     →(secx+tanx)(secx-tanx)=1

          secx-tanx=1/k    →  (2)          [From (1)]

Add (1) and (2)

secx+tanx+secx-tanx = k + 1/k

2secx = (k²+1)/k

 cosx = 2k/(k²+1)                  [secx = 1/cosx]

From identity sin²x+cos²x=1

      →cos²x=1-sin²x

         cosx = √(1-sin²x)

Then,

√(1-sin²x) = 2k(k²+1)

  1-sin²x = [2k/(k²+1)]²

  sin²x=1² - [2k/(k²+1)]²

  sin²x = [(k²+1+2k)/(k²+1)] [(k²+1-2k)/(k²+1)]

  sin²x = (k+1)²×(k-1)²/(k²+1)²

  sinx = (k+1)(k-1)/(k²+1)

  ∴sinx = (k²-1)/(k²+1)

In the above steps the identity (a-b)(a+b)=a²-b² is used a lot don't be confused.