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Mrac [35]
2 years ago
9

Proving trigonometric identities 2(cosx sinx-sinx cos2x)/sin2x =secx

Mathematics
1 answer:
eduard2 years ago
3 0

This is not an identity.

\dfrac{2(\cos(x)\sin(x) - \sin(x)\cos(2x))}{\sin(2x)} \neq \sec(x)

Check x = π/4, for which we have cos(π/4) = sin(π/4) = 1/√2. Together with sin(2•π/4) = sin(π/2) = 1 and cos(2•π/4) = cos(π/2) = 0, the left side becomes 1, while sec(π/4) = 1/cos(π/4) = √2.

Keeping the left side unchanged, the correct identity would be

\dfrac{2(\cos(x)\sin(x) - \sin(x)\cos(2x))}{\sin(2x)} = -2\cos(x) + 1 + \sec(x)

To show this, recall

• sin(2x) = 2 sin(x) cos(x)

• cos(2x) = cos²(x) - sin²(x)

• cos²(x) + sin²(x) = 1

Then we have

\dfrac{2(\cos(x)\sin(x) - \sin(x)\cos(2x))}{\sin(2x)} = \dfrac{2\cos(x)\sin(x) - 2\sin(x)\cos(2x)}{\sin(2x)} \\\\ = \dfrac{\sin(2x) - 2\sin(x)\cos(2x)}{\sin(2x)} \\\\ = 1 - \dfrac{2\sin(x)\cos(2x)}{\sin(2x)} \\\\ = 1 - \dfrac{2\sin(x)(\cos^2(x) - \sin^2(x))}{2 \sin(x)\cos(x)} \\\\ = 1 - \dfrac{\cos^2(x) - \sin^2(x)}{\cos(x)} \\\\ = 1 - \cos(x) + \dfrac{\sin^2(x)}{\cos(x)} \\\\ = 1 - \cos(x) + \dfrac{1 - \cos^2(x)}{\cos(x)} \\\\ = 1 - \cos(x) + \sec(x) - \cos(x) \\\\ = -2\cos(x) + 1 + \sec(x)

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Answer:

This equation has one solution

Step-by-step explanation:

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What is the GCF of 35,42,38
lbvjy [14]
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Please help me I need help today thank you and please show work
scoundrel [369]

17) x-8=11

18) 9-x=5

19) x+6=7

20) x+(-4)=-18

Step-by-step explanation:

17) The difference between a number and 8 is 11

Let number = x

then the equation will be:

x-8=11

Solving:

x-8=11

x=11+8\\x=19

18) Nine less than a number is five.

Let number = x

then the equation will be:

9-x=5

Solving:

9-x=5\\-x=5-9\\-x=-4\\x=4

19) Seven is equal to a number plus 6

Let number = x

then the equation will be:

x+6=7

Solving:

x+6=7\\x=7-6\\x=1

20) The sum of a number and -4 is -18

Let number = x

then the equation will be:

x+(-4)=-18

solving:

x+(-4)=-18\\x-4=-18\\x=-18+4\\x=-14

Keywords: Algebraic Expressions

Learn more about Algebraic Expressions at:

  • brainly.com/question/1617787
  • brainly.com/question/1600376
  • brainly.com/question/9720317

#learnwithBrainly

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7x + 2y = 4
Vinil7 [7]

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2 years ago
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