Correct Answer - Option 4 : None of these.
Concept:
In a right-angled triangle with length of the side opposite to angle θ as perpendicular (P), base (B) and hypotenuse (H):
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\(\rm\sin \theta =\frac{P}{H},\cos \theta =\frac{B}{H},\tan \theta =\frac{P}{B}\).
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\(\rm{{P}^{2}}+{{B}^{2}}={{H}^{2}}\) (Pythagoras' Theorem).
Trigonometric Ratios:
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csc θ = \(\rm \frac{1}{\sin \theta}\)
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sec θ = \(\rm \frac{1}{\cos \theta}\)
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tan θ = \(\rm \frac{\sin \theta}{\cos \theta}\)
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cot θ = \(\rm \frac{\cos \theta}{\sin \theta}\)
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cot θ = \(\rm \frac{1}{\tan \theta}\)
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Calculation:
Consider the given expression \(\rm 4\sin\theta-5\cos\theta\over4\sin\theta+5\cos\theta\).
Dividing the numerator and denominator by cos θ, we get:
= \(\rm 4\tan\theta-5\over4\tan\theta+5\)
Substituting the given value tan θ = \(4\over5\), we get:
= \(\rm 4\left({4\over5}\right)-5\over4\left({4\over5}\right)+5\)
= \(\rm 16-25\over16+25\)
= \(-9\over41\).
