$\frac{1+\sin x}{\cos x}+\frac{\cos x}{1+\sin x} \equiv \frac{2}{\cos x}$
Answer (1)
Find a common denominator for the left-hand side (LHS).
Expand and simplify the numerator using the Pythagorean identity sin 2 x + cos 2 x = 1 .
Factor and cancel common terms.
Verify that the simplified LHS is equal to the right-hand side (RHS): cos x 2 .
Explanation
Problem Analysis We are given the trigonometric identity to prove: cos x 1 + sin x + 1 + sin x cos x ≡ cos x 2 Our goal is to show that the left-hand side (LHS) is equal to the right-hand side (RHS).
Finding Common Denominator To prove the identity, we will start by combining the two fractions on the LHS. The common denominator will be the product of the individual denominators, which is cos x ( 1 + sin x ) .
Rewriting LHS Now, we rewrite the LHS with the common denominator: cos x ( 1 + sin x ) ( 1 + sin x ) 2 + cos 2 x
Expanding Numerator Next, we expand the numerator: cos x ( 1 + sin x ) 1 + 2 sin x + sin 2 x + cos 2 x
Using Pythagorean Identity We can use the Pythagorean identity sin 2 x + cos 2 x = 1 to simplify the numerator: cos x ( 1 + sin x ) 1 + 2 sin x + 1 = cos x ( 1 + sin x ) 2 + 2 sin x
Factoring Numerator Now, we factor out a 2 from the numerator: cos x ( 1 + sin x ) 2 ( 1 + sin x )
Canceling Common Factor We can cancel the common factor ( 1 + sin x ) from the numerator and the denominator: cos x 2
Conclusion Since the simplified LHS is equal to the RHS, the identity is proven.
Examples
Trigonometric identities are useful in various fields such as physics, engineering, and computer graphics. For example, they can be used to simplify complex expressions in wave mechanics, circuit analysis, and image processing. Understanding and proving trigonometric identities helps in simplifying calculations and solving problems in these areas.
