Question 4 Prove that: ( cos x - cos y )^2 + ( sin x - sin y )^2 = 4 [ sin (x-y) / 2 ]^2
Class X1 - Maths -Trigonometric Functions Page 82
Answers
Answered by
3
LHS = (cosx - cosy)² + (sinx - siny)²
use the formula,
cosC - cosD = 2.sin(C+D)/2.sin(D-C)/2
sinC - sinD = 2.cos(C+D)/2.sin(C-D)/2
= {2sin(x +y)/2.sin(y-x)/2}² + {2cos(x+y)/2.sin(x - y)/2}²
= 4{sin²(x + y)/2.sin²(x - y)/2} + cos²(x+y)/2.sin²(x - y).)/2}
= 4sin²(x-y)/2 [ sin²(x + y)/2 + cos²(x + y)/2] = 4sin²(x - y)/2 .1 [ use, sin²∅ + cos²∅=1]
= 4sin²(x-y)/2 = RHS
use the formula,
cosC - cosD = 2.sin(C+D)/2.sin(D-C)/2
sinC - sinD = 2.cos(C+D)/2.sin(C-D)/2
= {2sin(x +y)/2.sin(y-x)/2}² + {2cos(x+y)/2.sin(x - y)/2}²
= 4{sin²(x + y)/2.sin²(x - y)/2} + cos²(x+y)/2.sin²(x - y).)/2}
= 4sin²(x-y)/2 [ sin²(x + y)/2 + cos²(x + y)/2] = 4sin²(x - y)/2 .1 [ use, sin²∅ + cos²∅=1]
= 4sin²(x-y)/2 = RHS
Similar questions