Question

Simplify the trigonometric expression.

Answer 1

We can simplify the equation by taking the LCM, as shown below.
Note: I have used x in place of Ф as equation editor does not contain the symbol Ф. The steps will be same, regardless of the variable.


`(1)/(1+sin(x)) + (1)/(1-sin(x)) \\ \\ = (1-sin(x)+1+sin(x))/((1+sin(x))(1-sin(x))) \\ \\ = (2)/(1-sin^(2)(x) ) \\ \\ = (2)/(cos^(2)(x) ) \\ \\ =2sec^(2)(x)`

So, the answer to this questions is 2 sec²(Ф)

Answer 2

[1/(1+ sinθ)] + [1/(1- sinθ)]

= [(1 - sinθ) + (1 + sinθ)] / [(1+ sinθ)(1- sinθ)]

= (1 - sinθ + 1 + sinθ) / (1 - sinθ + sinθ - sin²θ)

= (1 + 1 - sinθ + sinθ) / (1 - sin²θ)

= 2/(1-sin²θ)