Respuesta :
Answer:
see explanation
Step-by-step explanation:
Using the addition identity for sine
sin(x + y) = sinxcosy - cosxsiny
Consider the left side
cos²(45 - A) - sin²(45 - A)
cos²(45 - A) = 1 - sin²(45 - A), thus
1 - sin²(45 - A) - sin²(45 - A)
= 1 - 2sin²(45 - A) ← expand sin(45 - A)
= 1 - 2(sin45cosA - cos45sinA)²
= 1 - 2([tex]\frac{\sqrt{2} }{2}[/tex]cosA - [tex]\frac{\sqrt{2} }{2}[/tex]sinA)²
= 1 - 2([tex]\frac{1}{2}[/tex]cos²A - sinAcosA + [tex]\frac{1}{2}[/tex]sin²A)
= 1 - cos²A + 2sinAcosA - sin²A
= sin²A + 2sinAcosA - sin²A
= 2sinAcosA
= sin2A = right side ⇒ verified
By Using the addition identity for sine; sin(x + y) = sinx cosy - cosx siny.
It is proved that cos²(45 - A) - sin²(45 - A)= Sin2A.
How to convert the sine of an angle to some angle of cosine?
We can use the fact that:
[tex]\sin(\theta ^\circ) = \cos(90 - \theta^\circ)[/tex]
To convert the sine to cosine (but the angles won't stay the same unless it's 45 degrees).
Using the addition identity for sine
sin(x + y) = sinx cosy - cosx siny
Now,
cos²(45 - A) - sin²(45 - A)
cos²(45 - A) = 1 - sin²(45 - A),
1 - sin²(45 - A) - sin²(45 - A)
= 1 - 2sin²(45 - A)
Expand sin(45 - A)
= 1 - 2(sin45cosA - cos45sinA)²
= 1 - 2(√2/2 cosA - √2/2 sinA)²
= 1 - 2(1/2 cos²A - sinAcosA + 1/2 sin²A)
= 1 - cos²A + 2sinAcosA - sin²A
= sin²A + 2sinAcosA - sin²A
= 2sinAcosA
cos²(45 - A) - sin²(45 - A) = sin2A = right side
Hence, it is verified.
Learn more about sine to cosine conversion here:
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