Chapter 7: Q. 106 (page 498)

If $z = \tan \frac{α}{2}$ , show that localid="1646416257885" $\cos α = \frac{1 - z^{2}}{1 + z^{2}}$ .

Short Answer

Expert verified

To show that $\cos α = \frac{1 - z^{2}}{1 + z^{2}}$ ,first use double angle and inverse trigonometry, and simplify the equation.

Step by step solution

Step 1. Given information.

Consider the given question,

$z = \tan \frac{α}{2}, \cos α = \frac{1 - z^{2}}{1 + z^{2}}$

Using the double-angle formula,

$\cos (2 θ) = 2 \cos^{2} θ - 1$

From inverse trigonometry,

$\cos (\cos^{- 1} x) = x$

Then,

$\tan^{- 1} = \cos^{- 1} (\frac{1}{\sqrt{1 + x^{2}}})$

Step 2. Simplify the equation.

Consider the given question,

$z = \tan \frac{α}{2} α = 2 \tan^{- 1} z α = 2 \cos^{- 1} (\frac{1}{\sqrt{1 + z^{2}}})$

Using the double-angle formula and inverse trigonometry,

$\cos α = \cos (2 \cos^{- 1} (\frac{1}{\sqrt{1 + z^{2}}})) = 2 \cos^{2} (2 \cos^{- 1} (\frac{1}{\sqrt{1 + z^{2}}})) - 1 = \frac{2}{1 + z^{2}} - 1 = \frac{1 - z^{2}}{1 + z^{2}}$

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If $z = \tan \frac{α}{2}$ , show that localid="1646416257885" $\cos α = \frac{1 - z^{2}}{1 + z^{2}}$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Simplify the equation.

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If z=tanα2, show that localid="1646416257885" cosα=1-z21+z2.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Simplify the equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

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Study anywhere. Anytime. Across all devices.

If $z = \tan \frac{α}{2}$ , show that localid="1646416257885" $\cos α = \frac{1 - z^{2}}{1 + z^{2}}$ .