Example 14 - Class 10

Transcript

Example 14 Prove that cot⁡〖𝐴 − cos⁡𝐴 〗/cot⁡〖𝐴 + cos⁡𝐴 〗 =(𝑐𝑜𝑠𝑒𝑐 𝐴 − 1)/(𝑐𝑜𝑠𝑒𝑐 𝐴 + 1) Taking L.H.S cot⁡〖𝐴 −〖 cos〗⁡𝐴 〗/cot⁡〖𝐴 +〖 cos〗⁡𝐴 〗 Writing everything in terms of sin A and cos A = (cos⁡〖 𝐴〗/sin⁡〖 𝐴〗 −〖 cos〗⁡〖 𝐴〗)/(cos⁡〖 𝐴〗/sin⁡〖 𝐴〗 +〖 cos〗⁡〖 𝐴〗 ) = cos⁡〖 𝐴 −〖 cos〗⁡〖 𝐴 sin⁡〖 𝐴〗 〗 〗/(sin⁡〖 𝐴〗/(cos⁡〖 𝐴 + cos⁡〖 𝐴 sin⁡〖 𝐴〗 〗 〗/sin⁡〖 𝐴〗 )) = ( (𝒄𝒐𝒔⁡〖 𝑨〗 − 𝒄𝒐𝒔⁡〖 𝑨〗 sin⁡〖 𝐴 〗))/((𝒄𝒐𝒔⁡〖 𝑨〗 + 𝒄𝒐𝒔⁡〖 𝑨〗 sin⁡〖 𝐴 〗)) = (𝒄𝒐𝒔⁡〖 𝑨〗 (1 − sin⁡〖 𝐴 〗))/(𝒄𝒐𝒔⁡〖 𝑨〗 (1 + sin⁡〖 𝐴 〗)) = ( (1 − sin⁡〖 𝐴 〗))/( (1 + sin⁡〖 𝐴 〗)) Dividing sin A on numerator and denominator = ( ((1 − sin⁡〖 𝐴 〗))/(𝒔𝒊𝒏 𝑨))/( ((1 + sin⁡〖 𝐴 〗))/(𝒔𝒊𝒏 𝑨 )) = ( 1/(𝑠𝑖𝑛 𝐴) − (𝑠𝑖𝑛 𝐴)/(𝑠𝑖𝑛 𝐴))/(1/(𝑠𝑖𝑛 𝐴) + (𝑠𝑖𝑛 𝐴)/(𝑠𝑖𝑛 𝐴)) = ( 1/(𝑠𝑖𝑛 𝐴) − 1)/(1/(𝑠𝑖𝑛 𝐴) + 1) = (𝑐𝑜𝑠𝑒𝑐 𝐴 − 1)/(𝑐𝑜𝑠𝑒𝑐 𝐴 + 1) = R.H.S. So, L.H.S = R.H.S Hence proved