How do I prove the following Identity?
sec x(sec x-cos x)=tan^2x
If there is a certain website or suggestion to help with these type of equations I would greatly appreciate it. I've been studying these for a while but still get pretty confused.
sec(x)*[sec(x)-cos(x)]=
sec(x)*sec(x)-sec(x)*cos(x)
Remark:
sec(x)=1/cos(x)
sec(x)*sec(x)-sec(x)*cos(x)=
1/cos(x) * 1/cos(x) - 1/cos(x) * cos(x)=
1/cos^2(x)-1=
1/cos^2(x) - cos^2(x)/cos^2(x)=
1-cos^2(x)/cos^2(x)
Remark:
sin^2(x)+cos^2(x)=1
sin^2(x)=1-cos^2(x)
1-cos^2(x)/cos^2(x) = sin^2(x)/cos^2(x) = tan^2(x)
LS = sec x(sec x-cos x)
= (1/cosx)(1/cosx - cosx)
= 1/cos^2x - 1
= sec^2 - 1
= tan^2x (by definition)
= RS
To prove the given identity sec x(sec x - cos x) = tan^2x, we need to simplify the left-hand side (LHS) expression until it matches the right-hand side (RHS) expression.
Let's start with the LHS expression: sec x(sec x - cos x).
Step 1: Expand the expression
Using the distributive property, we can expand sec x(sec x - cos x) as sec^2x - sec x*cos x.
Step 2: Convert sec to its reciprocal form using the identity sec x = 1/cos x
Using the reciprocal identity for secant, we can rewrite sec^2x as (1/cos x)^2 or 1/cos^2x.
Step 3: Substitute the new expression into the expanded form
Substituting the new expression 1/cos^2x into the expanded form, we get 1/cos^2x - sec x*cos x.
Step 4: Convert cos x to its reciprocal form using the identity cos x = 1/sec x
Using the reciprocal identity for cosine, we can rewrite cos x as 1/sec x.
Step 5: Substitute the new expression into the previous step
Substituting the new expression 1/sec x into the previous step, we get 1/cos^2x - (1/sec x)*(1/sec x).
Step 6: Simplify the expression further
To simplify 1/cos^2x - (1/sec x)*(1/sec x), we need to express sec x in terms of cos x.
Using the identity sec x = 1/cos x, we can rewrite sec x as 1/cos x. Now the expression becomes 1/cos^2x - (1/(1/cos x))*(1/(1/cos x)).
Simplifying further, we get 1/cos^2x - (cos x)^2/cos^2x.
Step 7: Combine the fractions
Using a common denominator, we can combine the fractions as (1 - cos^2x)/cos^2x.
Step 8: Apply the Pythagorean identity sin^2x + cos^2x = 1
Since sin^2x + cos^2x = 1, we can rewrite 1 - cos^2x as sin^2x.
The expression now simplifies to sin^2x/cos^2x.
Step 9: Apply the identity tan x = sin x/cos x
Using the identity tan x = sin x/cos x, we can replace sin^2x/cos^2x with (tan x)^2.
Therefore, the LHS expression becomes (tan x)^2, which matches the RHS of the given identity.
To help you further with similar equations, I recommend using online resources such as math forums, educational websites (e.g., Khan Academy), and textbooks on trigonometry. These resources provide step-by-step explanations, practice problems, and additional examples to improve your understanding of trigonometric identities.