Verified by Ajay Kumar
Simplify the expression: \(\frac{\tan\theta-\sec\theta+1}{\tan\theta+\sec\theta-1}\).
\(\frac{\sin\theta+\cos\theta+1}{\sin\theta}\)
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\(\frac{1-\sin\theta}{\cos\theta}\)
Answer: D
Solution: Substitute \(\tan\theta=\frac{\sin\theta}{\cos\theta}\) and \(\sec\theta=\frac1{\cos\theta}\). Then the expression becomes \(\frac{\sin\theta-1+\cos\theta}{\sin\theta+1-\cos\theta}\). Simplifying using standard trigonometric identities gives \(\frac{1-\sin\theta}{\cos\theta}\). Trick: Replace tan and sec with sine and cosine, then simplify the fraction.
व्यंजक को सरल कीजिए: \(\frac{\tan\theta-\sec\theta+1}{\tan\theta+\sec\theta-1}\)।
\(\frac{\sin\theta+\cos\theta+1}{\sin\theta}\)
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1
\(\frac{1-\sin\theta}{\cos\theta}\)
उत्तर: D
हल: \(\tan\theta=\frac{\sin\theta}{\cos\theta}\) और \(\sec\theta=\frac1{\cos\theta}\) रखें। तब व्यंजक \(\frac{\sin\theta-1+\cos\theta}{\sin\theta+1-\cos\theta}\) बन जाता है। मानक त्रिकोणमितीय सर्वसमिकाओं से सरल करने पर \(\frac{1-\sin\theta}{\cos\theta}\) मिलता है। ट्रिक: tan और sec को sine तथा cosine के रूप में बदलकर भिन्न को सरल करें।
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