Question

cosec theta+cot theta =q,cosec theta-cot theta =1/q,find sin theta and sec theta

Answer 1

Here is the answer to the asked question...Hope it helps.... please mark it as brainliest if it does help...

Answer 2

Add the given both.

$(\csc\theta+\cot\theta)+(\csc\theta-\cot\theta)=q+\dfrac{1}{q}\\\\\\\implies\ 2\csc\theta=\dfrac{q^2+1}{q}\\\\\\\implies\ \csc\theta=\dfrac{q^2+1}{2q}\\\\\\\implies\ \large\boxed{\sin\theta=\dfrac{2q}{q^2+1}}$

Subtract the second one from the first.

$(\csc\theta+\cot\theta)-(\csc\theta-\cot\theta)=q-\dfrac{1}{q}\\\\\\\implies\ &2\cot\theta=\dfrac{q^2-1}{q}\\\\\\\implies\ &\cot\theta=\dfrac{q^2-1}{2q}$

Now,

$\sec\theta\ =\ \dfrac{\csc\theta}{\cot\theta}\\\\\\\implies\ \sec\theta=\ \dfrac{\left(\dfrac{q^2+1}{2q}\right)}{\left(\dfrac{q^2-1}{2q}\right)}\\\\\\\implies\ \large\boxed{\sec\theta=\dfrac{q^2+1}{q^2-1}}$