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erastovalidia [21]
3 years ago
6

Can a irrational number be a rational number?

Mathematics
1 answer:
otez555 [7]3 years ago
3 0

No, an irrational number cannot be a rational number

If they could there would literally be no point in the term “irrational number” existing

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See Below.

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We want to verify the equation:

\displaystyle \frac{1}{1+\sin\theta} = \sec^2\theta - \sec\theta \tan\theta

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\displaystyle \frac{1}{1+\sin\theta}\left(\frac{1-\sin\theta}{1-\sin\theta}\right) = \sec^2\theta - \sec\theta \tan\theta

Simplify. The denominator uses the difference of two squares pattern:

\displaystyle \frac{1-\sin\theta}{\underbrace{1-\sin^2\theta}_{(a+b)(a-b)=a^2-b^2}} = \sec^2\theta - \sec\theta \tan\theta

Recall that sin²(θ) + cos²(θ) = 1. Hence, cos²(θ) = 1 - sin²(θ). Substitute:

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Split into two separate fractions:

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Rewrite the two fractions:

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By definition, 1 / cos(θ) = sec(θ) and sin(θ)/cos(θ) = tan(θ). Hence:

\displaystyle \sec^2\theta - \sec\theta\tan\theta \stackrel{\checkmark}{=}  \sec^2\theta - \sec\theta\tan\theta

Hence verified.

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