(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
Jim G.
Jul 16, 2018

#"see explanation"#

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"#

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