Cos x + cos y formula
Ex 3.3, 11 Important
Misc 3
Misc 4 Important
Ex 3.3, 14
Ex 3.3, 15
Misc 5
Misc 7 Important
Example 16 Important
Ex 3.3, 16 Important
Ex 3.3, 17
Ex 3.3, 18 Important
Ex 3.3, 19
Ex 3.3, 20
Ex 3.3, 21 Important You are here
Example 17 Important
Misc 6 Important
cos x + cos y formula
Last updated at April 16, 2024 by Teachoo
Ex 3.3, 21 Prove that (cos4𝑥 + cos3𝑥 + cos2𝑥)/(sin4𝑥 + sin3𝑥 + sin2𝑥 ) = cot 3x Solving L.H.S Solving Numerator and Denominator separately We know that cos x + cos y = 2cos ((𝑥 + 𝑦)/2) cos ((𝑥 −𝑦)/2) Replacing x by 4x and y by 2x cos 4x + cos 2x = 2cos ((4𝑥 + 2𝑥)/2). cos ((4𝑥 − 2𝑥)/2) = 2cos (6𝑥/2). cos (2𝑥/2) = 2 cos 3x . cos x Now cos 4x + cos 2x + cos 3x = 2cos 3x . cos x + cos 3x = cos 3x (2cos x + 1) Similarly, Solving denominator sin 4x + sin 2x + sin 3x We know that sin x + sin y = 2sin ((𝑥 + 𝑦)/2) sin ((𝑥 −𝑦)/2) Replacing x by 4x and y by 2x sin 4x + sin 2x = 2 sin ((4𝑥 +2𝑥)/2). cos ((4𝑥 − 2𝑥)/2) = 2 sin (6𝑥/2). cos (2𝑥/2) = 2 sin 3x . cos x Now, sin 4x + sin 2x + sin 3x = sin 4x + sin 2x + sin 3x = 2sin 3x . cos x + sin 3x = sin 3x (2cos x + 1) Hence, our equation becomes (cos4𝑥 + cos3𝑥 + cos2𝑥)/(sin4𝑥 + sin3𝑥 + sin2𝑥 ) = (cos3𝑥 (2 𝑐𝑜𝑠 𝑥 +1 ))/(sin3𝑥 (2 𝑐𝑜𝑠 𝑥 +1 )) = cos3𝑥/sin3𝑥 = cot 3x = R.H.S. Hence R.H.S. = L.H.S. Hence proved