Question

(cos0 + Sin0) (cot0 + tan0) = Sec0 + cosec0
​

Answer 1

To prove:

(cosθ + sinθ) (cotθ + tanθ) = secθ + cosecθ

Identities used:

• sin²θ + cos²θ = 1
• 1 ÷ sinθ = cosecθ
• 1 ÷ cosθ = secθ

Method:

$\bf{(cos \theta + sin \theta)(cot \theta + tan \theta)}\\\\\\= \sf{(cos \theta + sin \theta)\bigg(\dfrac{cos \theta}{sin \theta} + \dfrac{sin \theta}{cos \theta}\bigg)}$

$\rm{Taking \: LCM,}$

$\sf{(cos \theta + sin \theta) \bigg(\dfrac{cos^{2} \theta + sin^{2} \theta}{sin \theta \times cos \theta}\bigg)}$

$= \sf{(cos \theta + sin \theta)\bigg(\dfrac{1}{sin\theta . cos \theta}\bigg)}$

$= \sf{\dfrac{cos \theta + sin \theta}{sin \theta.cos \theta}}\\\\\\\\= \sf{\dfrac{cos \theta}{sin \theta.cos \theta}+ \dfrac{sin \theta}{sin \theta.cos \theta}}\\\\\\\\= \sf{\dfrac{1}{sin \theta}+ \dfrac{1}{cos \theta}}\\\\\\= \sf{cosec \theta + sec \theta }$

$= \boxed{\boxed{\red{\bf{sec \theta + cosec \theta}}}}\\\\\\\underline{\tt{Hence \: Proved}}$

Know more:

$\boxed{\begin{minipage}{6cm} Important Trigonometric identities :- \\ \\ \: \: 1)\:\sin^2\theta+\cos^2\theta=1 \\ \\ 2)\:\sin^2\theta= 1-\cos^2\theta \\ \\ 3)\:\cos^2\theta=1-\sin^2\theta \\ \\ 4)\:1+\cot^2\theta=\text{cosec}^2 \, \theta \\ \\5)\: \text{cosec}^2 \, \theta-\cot^2\theta =1 \\ \\ 6)\:\text{cosec}^2 \, \theta= 1+\cot^2\theta \\\ \\ 7)\:\sec^2\theta=1+\tan^2\theta \\ \\ 8)\:\sec^2\theta-\tan^2\theta=1 \\ \\ 9)\:\tan^2\theta=\sec^2\theta-1 \end{minipage}}$