Question

`(1+siny)/(cos y)+(cos y)/(1+sin y)`


`(cot x)/(tanx+cot x)=cos^2x`

csc^2θ·tan^2θ - 1 = tan^2θ

Answer 1

Explanation:

a)

`(1 + \sin y)/( \cos y) + ( \cos y)/(1 + \sin y) = (1 + 2 \sin y + \sin^(2) y + \cos ^(2) y)/( \cos y(1 + \sin y))`

`= (2 + 2 \sin y)/( \cos y (1+ \sin y)) = (2)/( \cos y) = 2 \sec y`

b)

`( \cot x)/( \tan x + \cot x) = ( \cos x)/( \sin x( ( \sin x)/( \cos x) + ( \cos x)/( \sin x) ))`

`= ( \cos x)/( ( \sin^(2) )/( \cos x) + \cos x ) = ( \cos x)/( ( \sin ^(2) x + \cos^(2) x)/( \cos x) )`

`= \cos ^(2) x`

c) Note that cscx = 1/sinx so

`\csc^(2) \theta \tan^(2) \theta - 1= (1)/( \cos^(2) \theta ) - 1`

`= (1 - \cos^(2) \theta )/( \cos \theta) = ( \sin^(2) \theta)/( \cos ^(2) \theta ) = \tan^(2) \theta`