Question

How to verify frac(sec theta/ csc theta- cot theta -frac(sectheta/csc theta +cot theta = 2csc theta

Answer 1

`(sec \theta)/( csc \theta- cot \theta )-(sec \theta)/(csc \theta +cot \theta)= 2csc`



`\ sec(x)= (1)/(cos(x)) \\ csc(x)= (1)/(sin(x)) \\ cot(x)=(sin(x))/(sin(x)) \\ sin^2(x)+cos^2(x)=1`



`(sec \theta( csc \theta+cot \theta))/( (csc \theta- cot \theta)( csc \theta+ cot \theta) )-(sec \theta( csc \theta- cot \theta))/(( csc \theta- cot \theta)(csc \theta +cot \theta)) \\ =(sec \theta csc \theta+sec \theta cot \theta-sec \theta csc \theta+sec \theta cot \theta)/( csc^2 \theta- cot^2 \theta )`

`=(2sec \theta cot \theta)/( csc^2 \theta- cot^2 \theta ) \\ = \frac{2 * (1)/(cos\theta)*\frac{cos\theta } {sin\theta }}{(\frac{1 } {sin\theta })^2-(\frac{cos\theta } {sin\theta })^2} \\ = \frac{\frac{2 } {sin\theta }}{\frac{1-cos^2\theta } {sin^2\theta }} \\ =\frac{\frac{2 } {sin\theta }}{\frac{sin^2\theta } {sin^2\theta }} \\ =\frac{2 } {sin\theta } \\ =2 * (1)/(sin\theta ) \\ =2csc \theta`



`(sec \theta)/( csc \theta- cot \theta )-(sec \theta)/(csc \theta +cot \theta)= 2csc`