Simplifying the Trigonometric Expression
We are given the expression \(\text{cosec } 79^\circ (1 - \cos 79^\circ)\) and the substitution \(t = \cos 79^\circ\). Our goal is to simplify this expression using the given substitution.
Step-by-Step Solution
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Substitute the given value of t:
Given \(t = \cos 79^\circ\), substitute this into the expression:
Expression = \(\text{cosec } 79^\circ (1 - t)\)
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Express cosecant in terms of sine:
Recall the trigonometric identity \(\text{cosec } \theta = \frac{1}{\sin \theta}\). Applying this:
Expression = \(\frac{1}{\sin 79^\circ} (1 - t)\)
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Relate sine and cosine using the Pythagorean identity:
We know that \(\sin^2 \theta + \cos^2 \theta = 1\). Therefore, \(\sin \theta = \sqrt{1 - \cos^2 \theta}\) (since \(79^\circ\) is in the first quadrant, where sine is positive).
Substituting \(\cos 79^\circ = t\), we get:
\(\sin 79^\circ = \sqrt{1 - (\cos 79^\circ)^2} = \sqrt{1 - t^2}\)
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Substitute sine back into the expression:
Now, substitute the expression for \(\sin 79^\circ\) back into our simplified expression:
Expression = \(\frac{1}{\sqrt{1 - t^2}} (1 - t) = \frac{1 - t}{\sqrt{1 - t^2}}\)
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Simplify the algebraic expression:
We can factor the term \(\sqrt{1 - t^2}\) using the difference of squares formula: \(1 - t^2 = (1 - t)(1 + t)\). So, \(\sqrt{1 - t^2} = \sqrt{(1 - t)(1 + t)}\).
The expression becomes:
Expression = \(\frac{1 - t}{\sqrt{(1 - t)(1 + t)}}\)
We can rewrite the numerator \((1 - t)\) as \((\sqrt{1 - t})^2\). This gives:
Expression = \(\frac{(\sqrt{1 - t})^2}{\sqrt{1 - t} \sqrt{1 + t}}\)
Cancel out one \(\sqrt{1 - t}\) term from the numerator and denominator:
Expression = \(\frac{\sqrt{1 - t}}{\sqrt{1 + t}}\)
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Combine into a single square root:
Using the property \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\), we get:
Expression = \(\sqrt{\frac{1 - t}{1 + t}}\)
Final Answer Verification
The simplified expression is \(\sqrt{\frac{1-t}{1+t}}\), which matches one of the given options.
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