Ex 3.3, 16 - Chapter 3 Class 11 Trigonometric Functions

Transcript

Ex 3.3, 16 Prove that 𝑐𝑜𝑠⁡〖9𝑥 −〖 𝑐𝑜𝑠〗⁡5𝑥 〗/(𝑠𝑖𝑛 17𝑥 − 𝑠𝑖𝑛⁡3𝑥 ) =−𝑠𝑖𝑛⁡〖2𝑥 〗/𝑐𝑜𝑠⁡10𝑥 Solving L.H.S 𝑐𝑜𝑠⁡〖9𝑥 −〖 𝑐𝑜𝑠〗⁡5𝑥 〗/(𝑠𝑖𝑛 17𝑥 − 𝑠𝑖𝑛⁡3𝑥 ) We solve cos 9x – cos 5x & sin 17x – sin 3x seperately cos 9x – cos 5x = – 2 sin ((9x+5x)/2) sin((9x−5x)/2) = – 2 sin (14𝑥/2) sin (4𝑥/2) = – 2 sin 7x sin (2x) sin 17x – sin 3x = 2 cos ((17x+3x)/2) sin((17x−3x)/2) = 2 cos (20𝑥/2) sin (14𝑥/2) = 2 cos 10x sin 7x Now, 𝑐𝑜𝑠⁡〖9𝑥 −〖 𝑐𝑜𝑠〗⁡5𝑥 〗/(𝑠𝑖𝑛 17𝑥 − 𝑠𝑖𝑛⁡3𝑥 ) = (−𝟐 〖𝐬𝐢𝐧 〗⁡〖(𝟕𝐱)〖𝐬𝐢𝐧 〗⁡〖(𝟐𝐱)〗 〗)/(𝟐 𝒄𝒐𝒔⁡〖(𝟏𝟎𝐱)𝐬𝐢𝐧⁡〖 (𝟕𝐱)〗 〗 ) = 〖−sin〗⁡〖(2x)〗/𝑐𝑜𝑠⁡〖(10x)〗 = R.H.S So, L.H.S. = R.H.S. Hence proved