Question

how do you prove?: (1+cscA)/secA = cosA + cotA

Answer 1

what's the definition of csc and sec ?
rewrite csc and sec in terms of sin cos.

Answer 2

(1/(1/sin))/(1/cos) = cos + cos/sin

Answer 3

when we have to verify the identity (prove) better way to do it is working on only one side
either left or right
our job is to prove L.H.S=R.H.S

Answer 4

\(\color{#000000 }{ \displaystyle \frac{ 1+\csc z }{\sec z} =\cos z+\cot z }\)

that's your problem,
And the simplest way is to recognize that;

\(\color{#000000 }{ \displaystyle \cos z=\frac{1}{\sec z} }\)

\(\color{#000000 }{ \displaystyle \cot z=\frac{\cos z}{\sin z}=\cos z\csc z =\frac{\csc z}{\sec z} }\)

Answer 5

so i suggest you to solve left side to prove its equal to right

Answer 6

and don't forget the plus sign `1+cscA`
so
\[\large\rm \color{Red}{\frac{1+cscA}{secA}} =cosA+cotA\]
\[\large\rm \color{Red}{\frac{1+\frac{1}{sinA}}{\frac{1}{cosA}}} =cosA+cotA\]
now just simplify the fraction

Answer 7

make sense ?

Answer 8

yes, thank you

Answer 9

did you get the same thing at left sidE ?
both sides are equal ?