Example 13 - Prove sin (x+y) / sin (x-y) = tan x + tan y

Example 13 Prove that sin⁡〖(𝑥+𝑦) 〗/sin⁡〖(𝑥+𝑦) 〗 = tan⁡〖𝑥 + tan⁡𝑦 〗/tan⁡〖𝑥 −tan⁡𝑦 〗 Taking L.H.S. sin⁡〖(𝑥+𝑦) 〗/sin⁡〖(𝑥+𝑦) 〗 = (𝐬𝐢𝐧⁡〖 𝒙 〗 𝐜𝐨𝐬⁡〖𝒚 + 𝐜𝐨𝐬⁡〖𝒙 𝐬𝐢𝐧⁡〖𝒚 〗〗〗)/𝐬𝐢𝐧⁡〖𝒙 𝐜𝐨𝐬⁡〖𝒚 − 𝐜𝐨𝐬⁡〖𝒙 𝐬𝐢𝐧⁡𝒚 〗〗〗 Dividing the numerator and denominator by 𝒄𝒐𝒔⁡𝒙 𝒄𝒐𝒔⁡𝒚, = ((sin⁡𝑥 cos⁡〖𝑦 + cos⁡〖𝑥 sin⁡𝑦 〗〗)/cos⁡〖𝑥 cos⁡𝑦 〗 )/((sin⁡𝑥 cos⁡〖𝑦 − cos⁡〖𝑥 sin⁡𝑦 〗〗)/cos⁡〖𝑥 cos⁡𝑦 〗 ) = ((sin⁡𝑥 cos⁡𝑦)/(cos⁡𝑥 cos⁡𝑦 ) + 〖cos 𝑥 sin〗⁡〖𝑦 〗/(cos⁡𝑥 cos⁡𝑦 ))/(sin⁡〖𝑥 cos⁡𝑦 〗/(cos⁡𝑥 cos⁡𝑦 ) − 〖𝑐𝑜𝑠 𝑥 sin〗⁡𝑦/〖cos 𝑥 cos〗⁡𝑦 ) = (sin⁡𝑥/cos⁡𝑥 + sin⁡〖𝑦 〗/cos⁡𝑥 )/(sin⁡𝑥/(cos⁡𝑥 ) − sin⁡𝑦/cos⁡𝑦 ) = 𝐭𝐚𝐧⁡〖𝒙 + 𝐭𝐚𝐧⁡𝒚 〗/𝐭𝐚𝐧⁡〖𝒙 − 𝐭𝐚𝐧⁡𝒚 〗

Example 13 - Chapter 3 Class 11 Trigonometric Functions

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