KCET · Maths · Trigonometric Ratios & Identities
\((\sin \theta+\cos \theta)(\tan \theta+\cot \theta)\) is equal to
- A \(\sin \theta \cos \theta\)
- B 1
- C \(\sec \theta+\operatorname{cosec} \theta\)
- D \(\sec \theta \operatorname{cosec} \theta\)
Answer & Solution
Correct Answer
(C) \(\sec \theta+\operatorname{cosec} \theta\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned}(\sin \theta+&\cos \theta)(\tan \theta+\cot \theta) \\ &=(\sin \theta+\cos \theta)\left[\frac{\sin \theta}{\cos \theta}+\frac{\cos \theta}{\sin \theta}\right] \\ &=(\sin \theta+\cos \theta)\left(\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\sin \theta \cdot \cos \theta}\right) \\ &=\frac{(\sin \theta+\cos \theta)}{\sin \theta \cdot \cos \theta}=\frac{1}{\cos \theta}+\frac{1}{\sin \theta} \\ &=\sec \theta+\operatorname{cosec} \theta \end{aligned}\)
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