(sin x + cos x)^2 tan x = tan x + 2 sin^2 x
please help me verify the identity
using trig identities and I need to verify please help
(sinx+cosx)^2 = sin^2x + 2sinx cosx + cos^2x = 1+2sinx cosx
Now you just take the last step, recalling that tanx = sinx/cosx
To verify the given trigonometric identity, we can simplify both sides of the equation and show that they are equal. Let's start by simplifying each side step by step:
Starting with the left-hand side (LHS):
(sin x + cos x)^2 tan x
Using the formula for expanding the square of a binomial:
(sin x + cos x)^2 = sin^2 x + 2sin x cos x + cos^2 x
Now we can substitute this expression back into the LHS of the equation:
(sin^2 x + 2sin x cos x + cos^2 x) tan x
Next, we distribute the tan x to each term inside the parenthesis:
sin^2 x tan x + 2sin x cos x tan x + cos^2 x tan x
Moving to the right-hand side (RHS):
tan x + 2 sin^2 x
Now that we have simplified both sides, let's compare them:
LHS: sin^2 x tan x + 2sin x cos x tan x + cos^2 x tan x
RHS: tan x + 2 sin^2 x
Since we want to show that LHS = RHS, we need to simplify the LHS further and see if it matches the RHS.
Factoring out a tan x from the LHS:
tan x (sin^2 x + 2sin x cos x + cos^2 x)
Using the trigonometric identity sin^2 x + cos^2 x = 1:
tan x (1 + 2sin x cos x)
Now we can see that the LHS matches the RHS:
tan x (1 + 2sin x cos x) = tan x + 2 sin^2 x
Therefore, the given identity (sin x + cos x)^2 tan x = tan x + 2 sin^2 x is verified.