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Let’s check out your problem:

(i) f(t)=(1+sqrtt)/(1-sqrtt)

(i) f(t)=1+t1−t f(t)=\frac{1+\sqrt{t}}{1-\sqrt{t}}

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Csc, sec, and cot of special anglesright chevron icon

Full solution

Q. (i) f(t)=1+t1−t f(t)=\frac{1+\sqrt{t}}{1-\sqrt{t}}
  1. Multiply by Conjugate: First, let's multiply the numerator and denominator by the conjugate of the denominator to rationalize it.\newlinef(t)=1+t1−t⋅1+t1+tf(t) = \frac{1+\sqrt{t}}{1-\sqrt{t}} \cdot \frac{1+\sqrt{t}}{1+\sqrt{t}}
  2. Perform Multiplication: Now, let's perform the multiplication in the numerator and the denominator.\newlineNumerator: (1+t)×(1+t)=1+2t+t(1+\sqrt{t}) \times (1+\sqrt{t}) = 1 + 2\sqrt{t} + t\newlineDenominator: (1−t)×(1+t)=1−t(1-\sqrt{t}) \times (1+\sqrt{t}) = 1 - t
  3. Final Rationalized Form: So, we have f(t)=1+2t+t1−tf(t) = \frac{1 + 2\sqrt{t} + t}{1 - t}
  4. Correct Denominator: But wait, there's a mistake in the denominator. It should be 1−(t)21 - (\sqrt{t})^2, which simplifies to 1−t1 - t. So, the correct denominator is 1−t1 - t, which I previously wrote correctly, but I doubted myself.

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