Verify the identity: cos x + sin x tan x = sec x.
LS = cosx + sinx(sinx/cosx)
= (cos^2x + sin^2x)/cosx
= 1/cosx
= sec x
= RS
To verify the identity cos x + sin x tan x = sec x, we need to manipulate the left side of the equation to show that it is equal to the right side (sec x).
First, let's express tan x in terms of sin x and cos x:
tan x = sin x / cos x
Now, substitute this expression into the left side of the equation:
cos x + sin x * (sin x / cos x)
Next, simplify the expression:
cos x + sin²x / cos x
To combine the terms, we need a common denominator. The common denominator will be cos x:
(cos x * cos x) / cos x + (sin²x) / cos x
Simplifying further, we get:
cos²x / cos x + sin²x / cos x
Using the identity sin²x + cos²x = 1, we substitute this in:
1 / cos x
Now, recall that sec x is equal to 1 / cos x. Hence, the left side is equal to sec x.
Therefore, the identity cos x + sin x tan x = sec x holds true.