Proving Identities: Compound Angle, Double Angle, Quotient & Reciprocal

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In summary, To prove the given identities, you can manipulate the left hand side of each equation by using compound angle formulas, double angle formulas, quotient identities, and reciprocal identities. For part a), you can start by dividing the numerator and denominator by cosx.
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Homework Statement



Prove the following identities.

a) cosx/1-sinx = secx + tanx

b) cos^2x+ sinxcosx/tanx = 2cos^2x


The Attempt at a Solution



Well what I tried doing was substituting the appropriate compound angle formulas, double angle formulas, quotient identities, and reciprocal identities, but I just can't seem to solve it all the way (if that's what "prove" means in the first place anyways)

Anyone have any idea how I'd go about solving them? This is what I have so far.

a) cosx/1-sinx = 1/cosx + sinx/cosx

b) I haven't started yet :/
 
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  • #2
to prove part a) you should start with one side of the equation and manipulate it to arrive at the other side of the equation.

I'd begin with the left hand side. To get started, divide numerator and denominator by a certain term (do you see which term?)
 

Related to Proving Identities: Compound Angle, Double Angle, Quotient & Reciprocal

1. What are identities in mathematics?

An identity in mathematics is a statement that is true for all values of the variables involved. It is a way to express a relationship between different mathematical expressions.

2. How are compound angle identities used?

Compound angle identities are used to simplify complex trigonometric expressions by expressing them in terms of simpler trigonometric functions. They are also useful in solving trigonometric equations and proving trigonometric identities.

3. What is the difference between double angle and compound angle identities?

The main difference between double angle and compound angle identities is that double angle identities involve a single angle while compound angle identities involve two or more angles. Double angle identities are used to express a trigonometric function of twice an angle, while compound angle identities involve adding, subtracting, or multiplying two or more trigonometric functions.

4. How can I prove a trigonometric identity?

To prove a trigonometric identity, you must manipulate one side of the equation using various trigonometric identities and algebraic properties until it is equivalent to the other side of the equation. This can involve expanding, factoring, and simplifying expressions, as well as using substitution and other trigonometric identities.

5. Why are identities important in mathematics?

Identities are important in mathematics because they provide a way to simplify complex expressions, solve equations, and prove other mathematical statements. They are also used in various applications, such as in physics and engineering, where trigonometric functions are heavily used.

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