Calculus: Double Angle & Power-Reducing Formulae (WNM) 3 Verify the trig identities by showing clear steps and use of relevant fundamental trig identities: a) - In(1 + cos x) = In(1-cos x) - 2In sin x (b) In(sin 0 tan 0) = 2In sin 0|-In(cos() 3. Without using a calculator, find all solutions of 4 sin' x + 2sin? x = 2 sin x+1 in the interval [0, 2x) .alculus: Double Angle & Power-Reducing Formulae (WNM) specified interval. with the aid of a calculator, sketch the graph 4 sin' x + 2 sin x = 2 sin x +1 and find all the solutions in theWe will derive the half angle identities by the Pythagorean identity and by the Power-Reducing formulas. See Proofs below: Method 1: By the Pythagorean Identity: cos? (+ sin? ( 2) =1 sin ? (2) =1-cost (2 ) sin = =1 1-cos 2; sub cos ' (0 ) = 1+ cos 20 2 1 + cos 2 sin =+1/1 2 2-1-cos0 sin = + 2 - COs O sir + 2 1-cos 2 A Method 2: By the Power-Reducing formulas: sin ( A) = 2 sin ( A) =+ 1-cos 2 A 2 let 2A =0= A = N / C 1-cos sin N /D 2 substituting : AF - cos sin 2 : What is the significance of in the formulas? Practice 1: Prove 1+ cos 0 COs : + (a) N 2 cos ( 2 ) let ? = APrecalculus: Half-Angle Formulas (WNM) 1- cos 0 ( b ) tan - = + 1+ cos0 : tan- 1-cos 0 (c) 2 sin 8 : sin 0 (d) 2 1+ cos0 : Example 1: Given that 0 = 240, use half-angle formulas to find the exact values of: 0 _ 240 Solution Note: 2 2 =120 = ~ is a Quadrant II angle cos 240 = - N / sin 1- cos 240 1-(-1/2) 1 0] (i) 2 4 2 COS (ii) tan (iii) CSC (iv) sec (v) cot - (vi) Reference: Ron Larson (2012): Precalculus with Limits - A Graphing Approac