Prove that # 1/(csctheta+1) + 1/(csctheta-1) -= 2sec theta \ tan theta #?
1 Answer
The identity as quoted is invalid, However:
# 1/(csctheta+1) + 1/(csctheta-1) -= 2sec theta \ tan theta #
Explanation:
The identity as quoted is invalid, However:
We have:
# 1/(csctheta+1) + 1/(csctheta-1) -= ((csctheta-1) + (csctheta+1))/((csctheta+1)(csctheta-1)) #
# " " = (csctheta-1 + csctheta+1)/(csc^2theta+csctheta-csctheta-1) #
# " " = (2csctheta)/(csc^2theta-1) #
Using the trig identity
# 1/(csctheta+1) + 1/(csctheta-1) = (2csctheta)/(1+cot^2theta-1) #
# " " = (2csctheta)/(cot^2theta) #
# " " = (2csctheta)(tan^2theta) #
# " " = (2/sintheta)(sin theta)/(cos theta)tan theta #
# " " = 2/(cos theta)tan theta #
# " " = 2sec theta \ tan theta #
