Question

Sin0(1+tan0)+cos0(1+cot0)=sec0+cosec0 prove this

Answer 1

Answer:

Hence Proved

Step-by-step explanation:

Sin0(1+tan0)+cos0(1+cot0)=sec0+cosec0 prove this

$Sin\theta(1+Tan\theta) + Cos\theta(1+Cot\theta)\\\\Tan\theta = \frac{Sin\theta}{Cos\theta} \: \& \:Cot\theta = \frac{Cos\theta}{Sin\theta}\\\\\implies Sin\theta(1+\frac{Sin\theta}{Cos\theta}) + Cos\theta(1+\frac{Cos\theta}{Sin\theta})\\\\\implies Sin\theta(\frac{Cos\theta + Sin\theta}{Cos\theta}) + Cos\theta(\frac{Sin\theta + Cos\theta}{Sin\theta})$

$\implies (Cos\theta + Sin\theta)(\frac{Sin\theta}{Cos\theta} + \frac{Cos\theta}{Sin\theta})\\\\\implies (Cos\theta + Sin\theta)(\frac{Sin^2\theta + Cos^2\theta}{Cos\theta Sin\theta})\\\\Sin^2\theta + Cos^2\theta = 1\\\\\implies (Cos\theta + Sin\theta)(\frac{1}{Cos\theta Sin\theta})$

$\implies \frac{Cos\theta}{Cos\theta Sin\theta} + \frac{Sin\theta}{Cos\theta Sin\theta}\\\\\implies \frac{1}{Sin\theta} + \frac{1}{Cos\theta}\\\\\implies Cosec\theta + Sec\theta\\\\= RHS$

QED

Answer 2

Hence proved  $\sin\theta(1+\tan\theta) + \cos\theta(1+\cot\theta)=\sec\theta+\csc\theta$

Step-by-step explanation:

To prove : $\sin\theta(1+\tan\theta) + \cos\theta(1+\cot\theta)=\sec\theta+\csc\theta$

Proof :

Taking LHS,

$LHS=\sin\theta(1+\tan\theta) + \cos\theta(1+\cot\theta)$

Writing tan and cot in terms of sin and cos,

$LHS=\sin\theta(1+\frac{\sin\theta}{\cos \theta}) + \cos\theta(1+\frac{\cos \theta}{\sin\theta})$

$LHS=\sin\theta(\frac{\cos\theta+\sin\theta}{\cos \theta}) + \cos\theta(\frac{\sin\theta+\cos \theta}{\sin\theta})$

$LHS=(\sin\theta+\cos\theta)(\frac{\sin\theta}{\cos\theta}+\frac{\cos \theta}{\sin\theta})$

$LHS=(\sin\theta+\cos\theta)(\frac{\sin^2\theta+\cos^2\theta}{\cos\theta\sin\theta})$

$LHS=(\sin\theta+\cos\theta)(\frac{1}{\cos\theta\sin\theta})$

$LHS=\frac{\sin\theta}{\cos\theta\sin\theta}+\frac{\cos\theta}{\cos\theta\sin\theta}$

$LHS=\frac{1}{\cos\theta}+\frac{1}{\sin\theta}$

$LHS=\sec\theta+\csc\theta$

$LHS=RHS$

Hence proved.

#Learn more

If sin0 +cos0 =√3 ,then prove that tan 0 + cot0 =1​

https://brainly.in/question/14321248