Rewrite the expression as a single logarithm and simplify th

Rewrite the expression as a single logarithm and simplify the result. 5 ln(cos^2 t) + 5 ln(l + tan^2 t) x Use the fundamental identities to simplify the expression. There is more than one correct form of the answer. cos^2 y/1 - sin y

Solution

1) 5 ln(cos²t) + 5 ln(1+tan²t)

= 5* [ ln(cos²t) + ln(1+tan²t)]   

[ use formula 1+tan²t = sec²t , ln (m) + ln (n) = ln ( m*n) , sec²t = 1/ cos²t ]

= 5 [ ln( cos²t * ( 1+tan²t) ]

= 5* [ ln( cos²t*sec²t)]

= 5*[ ln ( cos²t * ( 1/cos²t)

= 5*ln(1)

= 0

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2) cos² y / 1-sin y

= 1- sin² y / 1-sin y [ from sin²y + cos² y= 1]

= ( 1+sin y) ( 1- sin y) / ( 1- sin y) [ from a² - b² = (a+b) ( a-b) ]

= 1+ sin y