Q71 / 180 · KCET 2010 · Mathematics

$(\sin \theta+\cos \theta)(\tan \theta+\cot \theta)$ is equal to

  • A
    $\sin \theta \cos \theta$
  • B
    1
  • Correct answer
    C
    $\sec \theta+\operatorname{cosec} \theta$
  • D
    $\sec \theta \operatorname{cosec} \theta$

Solution

$\begin{aligned}(\sin \theta+&\cos \theta)(\tan \theta+\cot \theta) \\ &=(\sin \theta+\cos \theta)\left[\frac{\sin \theta}{\cos \theta}+\frac{\cos \theta}{\sin \theta}\right] \\ &=(\sin \theta+\cos \theta)\left(\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\sin \theta \cdot \cos \theta}\right) \\ &=\frac{(\sin \theta+\cos \theta)}{\sin \theta \cdot \cos \theta}=\frac{1}{\cos \theta}+\frac{1}{\sin \theta} \\ &=\sec \theta+\operatorname{cosec} \theta \end{aligned}$

Hence, the correct answer is option: 'C'.

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