To prove the trigonometric identity:
1/secx tanx = 1
we can follow these steps:
Break up the denominator of the left hand side as the product of two fractions:
1/secx tanx = 1/((1/cosx) (sin x/cos x)) = 1/(cosx/cosx) (sin x/cos x) = 1/(1) (sin x/cos x) = sin x/cos x
Rewrite each fraction property from the right side of the equation above as its equivalent identity:
sin x/cos x = sin x / (1/tan x) = sin x tan x
Rewrite both identities from part B in terms of only sign and cosine, using the identity:
sin x tan x = sin^2 x / cos x
Simplify the numerator of the right side of the equation above so there is only one squared term:
sin^2 x / cos x = (1 - cos^2 x) / cos x
Use the Pythagorean identity to rewrite the numerator of the right side of the equation. Remember to solve the identity for the appropriate quantity first:
(1 - cos^2 x) / cos x = (1 - cos^2 x) / (sqrt(1 - sin^2 x)) = (1 - (1 - sin^2 x)) / (sqrt(1 - sin^2 x)) = sin^2 x / (sqrt(1 - sin^2 x))
Separate the right side of the equation into two fractions around the minus sign:
sin^2 x / (sqrt(1 - sin^2 x)) = sin x / (sqrt(1 - sin^2 x)) - sin x / (sqrt(1 - sin^2 x))
Simplify the two fractions on the right side of the equation. Rewrite as:
sin^2 x / (sqrt(1 - sin^2 x)) = 2sin x / (2sqrt(1 - sin^2 x)) = sin x / sqrt(1 - sin^2 x)
The right side of the equation should match the given right side that was asked for in the proof. Therefore, the identity has been proved:
1/secx tanx = sin x / sqrt(1 - sin^2 x) = 1