How to prove sec x - tanx sinx= cosx
These are really easy.
Did you notice that for the ones I did for you, I changed everything to sines and cosines ?
LS = secx - tanxsinx
= 1/cosx -(sinx/cosx)sinx
= 1/cosx - sin^2 x/cosx
= (1 - sin^2 x)/cosx
= cos^2 x/cosx
= cosx = RS
for you its easy because you understand it !
To prove the identity sec(x) - tan(x) sin(x) = cos(x), we can start with the left-hand side (LHS) and simplify it step by step until we reach the right-hand side (RHS). Here's how it can be done:
Step 1: Start with the LHS of the equation:
sec(x) - tan(x) sin(x)
Step 2: Rewrite sec(x) as 1/cos(x) and tan(x) as sin(x)/cos(x):
1/cos(x) - sin(x)/cos(x) * sin(x)
Step 3: Find a common denominator by multiplying the two fractions:
(1 - sin^2(x))/cos(x)
Step 4: Remember the Pythagorean identity sin^2(x) + cos^2(x) = 1. Rearrange this identity to get: sin^2(x) = 1 - cos^2(x).
Substitute sin^2(x) = 1 - cos^2(x) in step 3:
(1 - (1 - cos^2(x)))/cos(x)
(1 - 1 + cos^2(x))/cos(x)
cos^2(x)/cos(x)
Step 5: Simplify by canceling out common factors in the numerator and denominator:
cos(x)
So, the LHS is equal to cos(x), which is the same as the RHS of the equation. Therefore, we have proven the identity sec(x) - tan(x) sin(x) = cos(x).