Question

please give the solution ​

Answer 1

Answer:

Step-by-step explanation:

We have,

$\tt{\dfrac{1+cot(\theta)}{sin(\theta)}+\dfrac{1+tan(\theta)}{cos(\theta)}}$

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

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

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

$\sf{=\{sin(\theta)+cos(\theta)\}\bigg\{\dfrac{1}{sin^2(\theta)}+\dfrac{1}{cos^2(\theta)}\bigg\}}$

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

$\sf{=\{sin(\theta)+cos(\theta)\}\bigg\{\dfrac{1}{sin^2(\theta)\,cos^2(\theta)}\bigg\}}$

$\sf{=\dfrac{sin(\theta)+cos(\theta)}{sin^2(\theta)\,cos^2(\theta)}}$

$\sf{=\dfrac{sin(\theta)}{sin^2(\theta)\,cos^2(\theta)}+\dfrac{cos(\theta)}{sin^2(\theta)\,cos^2(\theta)}}$

$\sf{=\dfrac{1}{sin(\theta)\,cos^2(\theta)}+\dfrac{1}{sin^2(\theta)\,cos(\theta)}}$

$\sf{=\dfrac{1}{sin(\theta)}\cdot\dfrac{1}{cos^2(\theta)}+\dfrac{1}{sin^2(\theta)}\cdot\dfrac{1}{cos(\theta)}}$

$\sf{=cosec(\theta)\cdot\,sec^2(\theta)+cosec^2(\theta)\cdot\,sec(\theta)}$

$\sf{=cosec(\theta)\,sec^2(\theta)+cosec^2(\theta)\,sec(\theta)}$