(a) how to write the product sin(7x) sin(3x) as a sum? (b)how to verify that tan(θ + π/4) tan(θ − π/4) = −1.?
1 Answer
Jul 16, 2018
Explanation:
#(a)#
#"using the "color(blue)"product to sum/difference formula"#
#•color(white)(x)2sinAsinB=cos(A-B)-cos(A+B)#
#"here "A=7x" and "B=3x#
#sin7xsin3x#
#=1/2(cos(7x-3x)-cos(7x+3x))#
#=1/2(cos4x-cos10x)#
#(b)#
#"using the "color(blue)"sum/difference identities for tan"#
#•color(white)(x)tan(x+-y)=(tanx+-tany)/(1∓tanxtany)#
#"note that "tan(pi/4)=1#
#tan(theta+pi/4)tan(theta-pi/4)#
#=(tantheta+tan(pi/4))/(1-tanthetatan(pi/4))xx(tantheta-tan(pi/4))/(1+tanthetatan(pi/4))#
#=(tantheta+1)/(1-tantheta)xx(tantheta-1)/(1+tantheta)#
#=(tan^2theta-1)/(1-tan^2theta)#
#=(-cancel((1-tan^2theta)))/cancel((1-tan^2theta))=-1" as required"#
