Misc 6 - Chapter 3 Class 11 Trigonometric Functions

Transcript

Misc 6 Prove that ((sin⁡〖7𝑥 + sin⁡〖5𝑥) + (sin⁡〖9𝑥 + sin⁡〖3𝑥)〗 〗 〗 〗)/((cos⁡〖7𝑥 + 𝑐𝑜𝑠 5𝑥) + (cos⁡〖9𝑥 + cos⁡〖3𝑥)〗 〗 〗 ) = tan 6x Solving L.H.S ((sin⁡〖7𝑥 + sin⁡〖5𝑥) + (sin⁡〖9𝑥 + sin⁡〖3𝑥)〗 〗 〗 〗)/((cos⁡〖7𝑥 + 𝑐𝑜𝑠 5𝑥) + (cos⁡〖9𝑥 + cos⁡〖3𝑥)〗 〗 〗 ) Lets solve numerator and Denominator separately Solving numerator (sin 7x + sin 5x) + ( sin 9x + sin 3x) = 2 sin ((7𝑥 + 5𝑥)/2) cos ((7𝑥 − 5𝑥)/2) + 2sin ((9𝑥 +3𝑥)/2) cos ((9𝑥 −3𝑥)/2) = 2 sin (12𝑥/2) . cos (2𝑥/2) + 2sin (12𝑥/2) cos (6𝑥/2) = 2 sin 6x . cos x + 2 sin 6x . cos 3x = 2 sin 6x (cos x + cos 3x) Now solving Denominator (cos 7x + cos 5x) + (cos 9x + cos 3x) = 2 cos ((7𝑥 + 5𝑥)/2) cos ((7𝑥 − 5𝑥)/2) + 2 cos ((9𝑥 +3𝑥)/2) cos ((9𝑥 −3𝑥)/2) = 2 cos (12𝑥/2) . cos (2𝑥/2) + 2 cos (12𝑥/2) cos (6𝑥/2) = 2 cos 6x . cos x + 2 cos 6x . cos 3x = 2 cos 6x (cos x + cos 3x) Now solving L.H.S ((sin⁡〖7𝑥 + sin⁡〖5𝑥) + (sin⁡〖9𝑥 + sin⁡〖3𝑥)〗 〗 〗 〗)/((cos⁡〖7𝑥 + 𝑐𝑜𝑠 5𝑥) + (cos⁡〖9𝑥 + cos⁡〖3𝑥)〗 〗 〗 ) = (𝟐 𝒔𝒊𝒏⁡𝟔𝒙 (𝒄𝒐𝒔⁡𝒙 + 𝒄𝒐𝒔⁡𝟑𝒙))/(𝟐 𝒄𝒐𝒔⁡𝟔𝒙 (𝒄𝒐𝒔⁡𝒙 + 𝒄𝒐𝒔⁡𝟑𝒙)) = (sin⁡6𝑥 )/(cos⁡6𝑥 ) = tan 6x = R.H.S Hence L.H.S = R.H.S Hence proved