Given that tan theta 34 pi Solution5 tan theta 34 And theta

Solution

5. tan theta =3/4    And theta is in third quadrant

Here opposite=3   and adjacent = 4

Hypotenuse = sqrt(9+16) = 5

And sin and cos are negative in third quadrant

sin theta = opposite/ hypotenuse = -3/5

cos theta=adjacent/hypotenuse = -4/5

sin 2 thea= 2 sin theta cos theta = 2(-3/5)(-4/5)=24/25

6. (1+cost)/(1-cost) - (1-cost)/(1+cost)=4cot t csct

Starting with left side

(1+ cos t)²-(1-cost)²/(1-cos²t) = (1+cost + 1-cos t)(1 + cos t -1 +cos t)/ sin²t

                                                    = 2*2 cos t/sin²t = 4 (cost/sint)(1/sint)= 4cot t csct