Question

If 3 cot θ 4 , show that`((1-tan^2theta))/((1+tan^2theta)) = (cos^2theta - sin^2theta)`

Answer 1

LHS = `((1-tan^2theta))/((1+tan^2theta))`

      =` ((1-1/cot^2theta))/((1+1/cot^2theta))`

     =`((cot^2theta-1)/(cot^2theta))/(((cot^2theta+1)/(cot^2theta)))`

     =`(cot^2theta-1)/(cot^2theta+1)`

    =`((4/3)^2-1)/((4/3)^2 +1)`          (𝐴𝑠, 3 cot 𝜃 = 4 𝑜𝑟 cot 𝜃 =`4/3`)

   =`(16/9-1)/(16/9+1)`

   =`(((16-9)/9))/(((16+9)/9))`

   =`((7/9))/((25/9))`

   = `7/25` 

𝑅𝐻𝑆 = `(cos^2 theta − sin^2 theta)`

  =`((cos^2theta - sin^2 theta))/1`

  =`(((cos^2theta-sin^2theta)/(sin^2theta)))/((1/(sin^2theta)))`

  =`((cos^2theta)/(sin^2theta)-(sin^2theta)/(sin^2theta))/(cosec^2theta)`

 =`((cot^2theta-1))/((cot^2theta+1))`

  =`([(4/3)^2-1])/([(4/3)^2+1])`

 =`((16/9-1/1))/((16/9+1/1))`

 =`(((16-9)/9))/(((16+9)/9))`

 =`((7/9))/((25/9))`

  =`7/25`

Since, LHS = RHS
Hence, verified.