2 questions !

tan theta = sin theta / cos theta

sin theta / cos theta = 1

sin theta = cos theta

If sin theta < 0, then cos theta must be equal to sin theta in order for tan theta to equal 1.

2 cos x = -1

cos x = -1/2

We would be looking at the 2nd quadrant...x has to be 120 degrees or 2pi/3.

for Q 5 ... so waht is the answer of the question !!! 

for Q 6 ... the answer in the model answer is   +and -  2pi/3

5. The sin is < 0 in quadrants 3 and 4. And the cos < 0 in quadrants 2 and 3. But the sin and the cos  would have to have the same sign for the tangent to be positive. Thus, theta is a 3rd quadrant angle. And since the tan(theta)= 1 at 45 degrees, this is a 45 degree angle in quadrant 3. And the cos of 45 degrees  = 1/√2, or √2/2, if you prefer. But, since the cos is negative in this quadrant, then cos(theta) = - 1/√2 = -√2/2.

6.  

2 cos x = -1   divide both sides by 2

cos x = -1/2    ....    this occurs at (2/3)pi in the interval [0, pi)  and at -(2/3)pi in the interval [-pi, 0]. The second answer may be a little hard to see. Note that cos x = -1/2 at (4/3)pi. But, subtracting 2*pi from this gives us -(2/3)pi.......

thank you CPhill  and AzizHusain