Misc 3 - Chapter 3 Class 11 Trigonometric Functions

Transcript

Misc 3 Prove that: (cos 𝑥 + cos y)2 + (sin 𝑥 – sin y)2 = 4cos2 (𝑥 + y)/2 Solving LHS (𝐜𝐨𝐬⁡𝒙+𝐜𝐨𝐬⁡𝒚 )^𝟐 + (𝐬𝐢𝐧⁡𝒙–𝐬𝐢𝐧⁡𝒚 )^𝟐 = cos2 𝑥+cos2 𝑦+2 cos⁡𝑥 cos⁡𝑦+sin2 𝑥+ sin2 𝑦 – 2sin⁡𝑥 sin⁡𝑦 = (𝒄𝒐𝒔𝟐 𝒙+𝒔𝒊𝒏𝟐 𝒙)+(𝒄𝒐𝒔𝟐 𝒚+𝒔𝒊𝒏𝟐 𝒚 )+2 (cos⁡𝑥 cos⁡𝑦 –sin⁡𝑥 sin⁡𝑦) = 1 + 1 +2 (𝐜𝐨𝐬⁡𝒙 𝐜𝐨𝐬⁡𝒚–𝐬𝐢𝐧⁡𝒙 𝐬𝐢𝐧⁡𝒚 ) = 2 + 2 𝒄𝒐𝒔⁡(𝒙 + 𝒚) = 2 [1 + 𝒄𝒐𝒔⁡(𝒙 + 𝒚) ] = 2 [1+2cos2 ((𝑥 + y)/2) – 1 ] = 4 cos^2 ((𝑥 + y)/2) = RH.S. Hence proved