Question

If tan theta = 3/4 then find the value of cos 2 theta - sin 2 theta.

Answer 1

 sin(θ) = 2/5 , 90° ≤ θ ≤ 180°. 
sin(β) = 1/2 , 0° ≤ β ≤ 90°. 

1) sin²(β) + cos²(β) = 1 
cos²(β) = 1 - sin²(β) 
= 1 - (1/2)² = 3/4 
cos(β) = ±½√3 . Note however that cos is positive on [0, 90> , so 

cos(β) = ½√3 

2) 
sec²(β) ≡ 1/cos²(β) = 4/3 

3) 
tan(θ) = sin(θ)/cos(θ) 
= sin(θ) / ( - √(1-sin²(θ) ) [minus sign because cos is negative on interval <90,180>] 
= (2/5) / ( - √( 1 - 4/25) ) 
= -2 / √21 
4) 
cosec(β) + tan(90-θ) = 
1/sin(β) + 1/tan(θ) = 
1/ (1/2) + (-(√21) /2) = 
2 - ½√21

Answer 2

hope it would help you...