Im having serious issues with verifying Trig. identities. Please help! Here is an example:
(tanX-secX)^2= 1-sinX/1+sinX
(sec x - tanx)^2 = (1-sinx)/(1+sinx)
use these substitutions
sin²(x) + cos²(x) = 1
cos²(x) = 1 - sin²(x)
(sec(x) - tan(x))²
=(1/cos(x) - sin(x)/cos(x))²
= (1-sin(x))² / cos²(x)
= (1-sin(x))² / (1 - sin²(x))
= (1-sin(x))² / [(1+sin(x))(1-sin(x))]
= (1-sin(x)) / (1+sin(x))
To verify the given trigonometric identity, let's start with the left-hand side (LHS) and simplify it:
Step 1: Expand and simplify the LHS:
(tanX - secX)^2 = (tanX - 1/cosX)^2
Using the identity tanX = sinX/cosX and secX = 1/cosX, we can rewrite it as:
(sinX/cosX - 1/cosX)^2
Combining the terms with a common denominator:
[(sinX - 1)/cosX]^2 = (sinX - 1)^2/cos^2X
Expanding and simplifying:
(sinX - 1)^2/cos^2X = sin^2X - 2sinX + 1 / cos^2X
Step 2: Simplify the right-hand side (RHS):
To simplify the right-hand side (RHS), we need to rationalize the denominator. Multiply the numerator and denominator by (1 - sinX) to do so:
[(1 - sinX) * (1 + sinX)] / [(1 + sinX) * (1 - sinX)]
Simplifying:
(1 - sin^2X) / (1 - sin^2X)
Using the identity 1 - sin^2X = cos^2X:
cos^2X / cos^2X = 1
Step 3: Compare LHS and RHS:
LHS = sin^2X - 2sinX + 1 / cos^2X
RHS = 1
As we can see, LHS = RHS, which verifies the given trigonometric identity.
When working with trigonometric identities, it's essential to familiarize yourself with the basic trigonometric ratios and identities. Additionally, using the fundamental Pythagorean identity (sin^2X + cos^2X = 1) and other algebraic manipulations can often help simplify and transform expressions to prove identities.