## anonymous one year ago iif cosectheta-cottheta=1\2 then costheta=?

• This Question is Open
1. welshfella

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

2. welshfella

* typo in line 6 it should be sqrt( 1 + cot^2 O) - cot O = 1/2

3. welshfella

you can also find the value of angle O from cot O then find cos O

4. welshfella

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

5. Loser66

$$csc \theta =\dfrac{1}{sin\theta}\\cot \theta=\dfrac{cos\theta}{sin\theta}$$ hence $$csc(\theta)-cot(\theta)= \dfrac{1}{sin(\theta)}-\dfrac{cos(\theta)}{sin(\theta)}=\dfrac{1-cos(\theta)}{sin(\theta)}=\dfrac{1}{2}$$

6. Loser66

Now, restrict the solution from 0, since if $$\theta =0$$ , then $$sin(\theta )=0$$, that make the expression undefined.

7. Loser66

we have: $$2(1-cos(\theta))= sin(\theta)$$ square both sides and solve for $$cos (\theta)$$