Prove identity (sinx+cosx)/(secx+cscx)= sinxcosx

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In summary, proving identity in mathematics means demonstrating that an equation is true for all values of the variable(s) involved. To prove the identity (sinx+cosx)/(secx+cscx)= sinxcosx, basic trigonometric identities such as sinx=1/cscx, cosx=1/secx, and sin^2x+cos^2x=1 can be used to manipulate both sides until they are equal to each other. The purpose of proving identities in trigonometry is to strengthen understanding and simplify complex expressions. There are no specific steps or rules to follow, but a strong understanding of basic trigonometric identities and properties and the ability to manipulate equations using algebraic rules is important. An
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guinessvolley
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prove this identity

(sinx+cosx)/(secx+cscx)= sinxcosx if you could list out the steps it would be appreciated
 
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Hello, and welcome to MHB! (Wave)

Since the RHS is in terms of the sine and cosine functions, the first thing I would do is write the LHS in terms of these functions only:

\(\displaystyle \frac{\sin(x)+\cos(x)}{\dfrac{1}{\cos(x)}+\dfrac{1}{\sin(x)}}\)

Now, combine terms in the denominator...what do you get?
 

What is the identity being asked to prove?

The identity being asked to prove is (sinx+cosx)/(secx+cscx)= sinxcosx.

What does the abbreviation "sin" stand for in this identity?

In this identity, "sin" stands for the sine function, which represents the ratio of the opposite side to the hypotenuse in a right triangle.

What does the abbreviation "cos" stand for in this identity?

In this identity, "cos" stands for the cosine function, which represents the ratio of the adjacent side to the hypotenuse in a right triangle.

What is the purpose of proving this identity?

Proving this identity is important in mathematics because it helps to establish the relationship between different trigonometric functions and demonstrates their equivalence.

What are the steps to proving this identity?

To prove this identity, we can use the fundamental trigonometric identities and algebraic manipulation. First, we can rewrite the left side of the equation using the reciprocal identities for secant and cosecant. Then, we can use the Pythagorean identities to simplify the expression further. Finally, we can use algebraic manipulation to show that the left side is equal to the right side of the equation.

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