Proving Trig Identities: Tan x Sec^4x = Tan x Sec^2x + Tan^3x Sec^2x

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In summary: Here it is.$$\tan x\sec^2x + \tan^3x\sec^2x$$$$\tan x (\tan^2 x + 1)\sec^2x$$$$\tan^3 x + \tan x\sec^2x$$ That's the same thing. Combine them.In summary, the goal is to show that $$\tan x\sec^4x\equiv\tan x\sec^2x + \tan^3x\sec^2x$$ and using trig identities/formulae, we can rewrite the right side as $$\tan x (\tan^2 x + 1)\sec^2x$$ and by
  • #1
trollcast
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Homework Statement



Show that:

$$\tan x\sec^4x\equiv\tan x\sec^2x + \tan^3x\sec^2x$$

Homework Equations



Trig identities / formulae

The Attempt at a Solution



I've got 2 different starts for it but I'm stuck after a few steps with both of them:

Attempt 1:

$$\tan x \sec^4 x$$
$$\frac{\tan x}{\cos^4 x}$$
$$\frac{\frac{\sin x}{\cos x}}{\cos^4 x}$$
$$\frac{\sin x \cos^4 x}{\cos x}$$
$$\sin x \cos^3 x$$

And then I can't think on anything else for this one.

Attempt 2:

$$\tan x \sec^4 x$$
$$\tan x (\tan^2 x + 1)^2$$
$$\tan x (\tan^4 x + 2\tan^2 x + 1)$$
$$\tan^5 x + 2\tan^3 x + \tan x$$

This one looks a bit closer since its got the higher power tans in it but I can't see where to get the sec terms from?
 
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  • #2
trollcast said:

Homework Statement



Show that:

$$\tan x\sec^4x\equiv\tan x\sec^2x + \tan^3x\sec^2x$$

Fastest way: split the second term up to ##\displaystyle \tan x\tan^2x\sec^2x##, then use the identity ##\displaystyle \tan^2x = \sec^2x - 1##.
 
  • #3
Curious3141 said:
Fastest way: split the second term up to ##\displaystyle \tan x\tan^2x\sec^2x##, then use the identity ##\displaystyle \tan^2x = \sec^2x - 1##.

How do you get that for the second term as there's nothing like it in our formula books?
 
  • #4
trollcast said:
How do you get that for the second term as there's nothing like it in our formula books?
If it's not in there, it should be. Certainly you know this one:
sin2x + cos2x = 1

If you divide both sides of this equation by cos2x, you get:
tan2x + 1 = sec2x

If you divide both sides of the first identity by sin2x, you get:
1 + cot2x = csc2x

You should have the first of these memorized. The latter two you can derive quickly.
 
  • #5
trollcast said:
How do you get that for the second term as there's nothing like it in our formula books?

##\displaystyle \sin^2x + \cos^2x = 1##. Divide throughout by ##\displaystyle \cos^2x##. Rearrange.

In fact, the tan-sec identify is a well-known one in its own right. So is the cot-cosec one, which you can derive by dividing the above equation by ##\displaystyle \sin^2x## instead of ##\displaystyle \cos^2x##.
 
  • #6
trollcast said:

Homework Statement



Show that:

$$\tan x\sec^4x\equiv\tan x\sec^2x + \tan^3x\sec^2x$$

Homework Equations



Trig identities / formulae

The Attempt at a Solution



I've got 2 different starts for it but I'm stuck after a few steps with both of them:

Attempt 1:

$$\tan x \sec^4 x$$
$$\frac{\tan x}{\cos^4 x}$$
$$\frac{\frac{\sin x}{\cos x}}{\cos^4 x}$$
$$\frac{\sin x \cos^4 x}{\cos x}$$
This is wrong. It should be
[tex]\frac{sin(x)}{cos^5(x)}[/tex]

$$\sin x \cos^3 x$$

And then I can't think on anything else for this one.

Attempt 2:

$$\tan x \sec^4 x$$
$$\tan x (\tan^2 x + 1)^2$$
$$\tan x (\tan^4 x + 2\tan^2 x + 1)$$
$$\tan^5 x + 2\tan^3 x + \tan x$$

This one looks a bit closer since its got the higher power tans in it but I can't see where to get the sec terms from?
Have you tried doing the same thing to the right side?
 

FAQ: Proving Trig Identities: Tan x Sec^4x = Tan x Sec^2x + Tan^3x Sec^2x

What is a "Trig proof"?

A "Trig proof" is a mathematical proof that uses trigonometric identities and properties to verify or prove a given statement or equation involving trigonometric functions.

What are some common trigonometric identities used in a Trig proof?

Some common trigonometric identities used in a Trig proof include the Pythagorean identities, sum and difference identities, double angle identities, and half angle identities.

How do you approach a Trig proof?

The best approach for a Trig proof is to start by identifying the given statement or equation and the desired outcome. Then, use known trigonometric identities and properties to manipulate the equation and simplify it until it matches the desired outcome.

What are some tips for solving a Trig proof?

Some tips for solving a Trig proof include being familiar with the basic trigonometric identities, practicing regularly, and breaking down the proof into smaller steps. It is also helpful to draw diagrams and use substitution to simplify the equations.

What are some common mistakes to avoid in a Trig proof?

Some common mistakes to avoid in a Trig proof include using incorrect trigonometric identities, not simplifying the equations enough, and making calculation errors. It is important to double-check all steps and ensure that the final solution matches the desired outcome.

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