iif cosectheta-cottheta=1\2 then costheta=?

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iif cosectheta-cottheta=1\2 then costheta=?

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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I'll use O for theta:- csc O - cot O = 1/2 what is cos O you might do it this way csc^2 O = 1 + cot^2 O so substituting ; sqrt( 1 + cot^2 O) + cot O = 1/2 solve for cot O then you can find cos O using other trig identities
* typo in line 6 it should be sqrt( 1 + cot^2 O) - cot O = 1/2
you can also find the value of angle O from cot O then find cos O

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I'll start the solution of the equation for you sqrt( 1 + cot^2 O) + cot O = 1/2 sqrt( 1 + cot^2 O) = cot O + 1/2 square both sides;- 1 + cot^2 O = cot^2 O + cot O + 1/4 now finding cot O is straightforward
\(csc \theta =\dfrac{1}{sin\theta}\\cot \theta=\dfrac{cos\theta}{sin\theta}\) hence \(csc(\theta)-cot(\theta)= \dfrac{1}{sin(\theta)}-\dfrac{cos(\theta)}{sin(\theta)}=\dfrac{1-cos(\theta)}{sin(\theta)}=\dfrac{1}{2}\)
Now, restrict the solution from 0, since if \(\theta =0\) , then \(sin(\theta )=0\), that make the expression undefined.
we have: \(2(1-cos(\theta))= sin(\theta)\) square both sides and solve for \(cos (\theta)\)

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