Question

cos? (45° + x) – sin’ (45º – x) is independent of x.​

Answer 1

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(\frac{\sqrt{2} }{2}

2

2

cosA - \frac{\sqrt{2} }{2}

2

2

sinA)²

= 1 - 2(\frac{1}{2}

2

1

cos²A - sinAcosA + \frac{1}{2}

2

1

sin²A)

= 1 - cos²A + 2sinAcosA - sin²A

= sin²A + 2sinAcosA - sin²A

= 2sinAcosA

= sin2A = right side ⇒ verified