Prove the identity # (cot x)/(1+csc x)-(cos x)/(1-csc x) -= secx + sinxtanx#?

1 Answer
Steve M
Jan 19, 2018

# (cot x)/(1+csc x)-(cos x)/(1-csc x) -= secx + sinxtanx#

Explanation:

We can manipulate the expression as follows:

# E = (cot x)/(1+csc x)-(cos x)/(1-csc x) #

# \ \ \ = ((cot x)(1-csc x)-(cos x)(1+csc x))/((1+csc x)(1-csc x) )#

# \ \ \ = ((cosx/sinx)(1-1/sinx)-(cos x)(1+1/sinx))/(1-csc^2)#

# \ \ \ = (cosx/sinx-cosx/sin^2x-cosx-cosx/sinx)/(-cot^2)#

# \ \ \ = (cosx/sin^2x+cosx)/(cot^2x)#

# \ \ \ = ((cosx+cosxsin^2x)/sin^2x)/(cos^2x/sin^2x)#

# \ \ \ = (cosx+cosxsin^2x)/(cos^2x)#

# \ \ \ = cosx/(cos^2x)+(cosxsin^2x)/(cos^2x)#

# \ \ \ = 1/(cosx)+(sin^2x)/(cosx)#

# \ \ \ = secx + sinxtanx#

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