Question

Sin (45+x) - sin (45-x) = 1/2 cos x​

Answer 1

Step-by-step explanation:

sin(45+x)·sin(45-x) = (1/2)cos2x

Use the sum and difference of two angles identity:

(sin45cosx+sinxcos45)·(sin45cosx-sinxcos45) = (1/2)cos2x

This is of the form (a+b)(a-b) = a2 - b2, so it's:

sin245cos2x - sin2xcos245 = (1/2)cos2x

Use The Pythagorean Identity sin2θ+cos2θ=1, so sin2θ=1-cos2θ. Also sin245 = cos245 = 1/2

(1/2)cos2x - (1-cos2x)(1/2) = (1/2)cos2x

(1/2)(cos2x - 1 + cos2x) = (1/2)cos2x

(1/2)(2cos2x - 1) = (1/2)cos2x

Last identity: cos2x = 2cos2x - 1:

(1/2)cos2x = (1/2)cos2x

here's your answer

Answer 2

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$\Large\fbox{\color{purple}{SOLITION}}$

Sin (45+x) - sin (45-x) = 1/2 cos x

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$\bf\Huge\red{\mid{\overline{\underline{ ANSWER }}}\mid }$

Use the sum and difference of two angles identity:    

(sin45cosx+sinxcos45)·(sin45cosx-sinxcos45) = (1/2)cos2x

This is of the form (a+b)(a-b) = a^2 - b^2, so it's:

     sin^245cos^2x - sin^2xcos^245 = (1/2)cos2x

 

 Use The Pythagorean Identity sin^2θ+cos^2θ=1, so sin^2θ=1-cos^2θ.  Also sin^245 = cos^245 = 1/2

     (1/2)cos^2x - (1-cos^2x)(1/2) = (1/2)cos2x

2x)(1/2) = (1/2)cos2x    

(1/2)(cos^2x - 1 + cos2x) = (1/2)cos2x

2x - 1 + cos2x) = (1/2)cos2x 

    (1/2)(2cos^2x - 1) = (1/2)cos2x

 

 Last identity: cos2x = 2cos^2x - 1:

2x - 1:  

   (1/2)cos2x = (1/2)cos2x

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