Cot2A+tanA=?

2 Answers

Answer:

#csc2A#

Explanation:

#cot2A+tanA#

#=(cos2A)/(sin2A)+sinA/cosA#

# =(cos2AcosA+sin2AsinA)/(sin2AcosA)#

# =cos(2A-A) /(sin2AcosA)#

# =cosA/(sin2AcosA)#

# =1/(sin2A) #

# =csc2A#

Formulae:

  • #cosCcosD+sinCsinD=cos(C-D)#
  • #cos2AcosA+sin2AsinA=cos(2A-A)#
Feb 24, 2018

Answer:

= csc 2A

Explanation:

Call tan A = t , and apply the trig identity
#tan 2A = (2tan A)/(1 - tan^2 A)#
The expression becomes:
#f (A) = cot 2A + tan A = (1 - t^2)/(2t) + t = ((1 - t^2) + 2t^2)/(2t)#
#f(A) = (1 + t^2)/(2t) = (1 +tan^2 A)/(2tan A)#
Using trig identity, replace #(1 + tan^2 A)# by #(sec^2 A)#, we get:
#f(A) = (sec^2 A)((2sin A)/(cos A)) = (cos A)/(2sin A.cos^2 A) #
Note that: #sin 2A = 2sin A.cos A# -->
#f(A) = 1/(sin 2A) = csc 2A#