Question

please help,Trigonometric maths question​

Answer 1

Answer:

Step-by-step explanation:

We know,

$\star\,\,\bold{\tt{cos(\theta+\phi)=cos(\theta)cos(\phi)-sin(\theta)sin(\phi)}}\\\star\,\,\bold{\tt{sin(\theta+\phi)=sin(\theta)cos(\phi)+cos(\theta)sin(\phi)}}$

Now,

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

$\sf{\implies\,cot(\theta+\phi)}=\dfrac{cos(\theta)cos(\phi)-sin(\theta)sin(\phi)}{sin(\theta)cos(\phi)+cos(\theta)sin(\phi)}}$

$\star\,\,\bold{\tt{Divide\,\,\,numerator\,\,\,and\,\,\,denominator\,\,\,by\,\,\,sin(\theta)sin(\phi)}}$

$\sf{\implies\,cot(\theta+\phi)}=\dfrac{\dfrac{cos(\theta)cos(\phi)-sin(\theta)sin(\phi)}{sin(\theta)sin(\phi)}}{\dfrac{sin(\theta)cos(\phi)+cos(\theta)sin(\phi)}{sin(\theta)sin(\phi)}}}$

$\sf{\implies\,cot(\theta+\phi)}=\dfrac{\dfrac{cos(\theta)cos(\phi)}{sin(\theta)sin(\phi)}-1}{\dfrac{sin(\theta)cos(\phi)}{sin(\theta)sin(\phi)}+\dfrac{cos(\theta)sin(\phi)}{sin(\theta)sin(\phi)}}}$

$\sf{\implies\,cot(\theta+\phi)}=\dfrac{\dfrac{cos(\theta)}{sin(\theta)}\cdot\dfrac{cos(\phi)}{sin(\phi)}-1}{\dfrac{cos(\phi)}{sin(\phi)}+\dfrac{cos(\theta)}{sin(\theta)}}}$

$\sf{\implies\,cot(\theta+\phi)}=\dfrac{cot(\theta)\cdot\,cot(\phi)-1}{cot(\phi)+cot(\theta)}}$

$\sf{\implies\,cot(\theta+\phi)}=\boxed{\dfrac{cot(\theta)cot(\phi)-1}{cot(\theta)+cot(\phi)}}}$