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Chapter 5: Problem 46

verify the identity. $$\sec ^{2}\left(\frac{\pi}{2}-x\right)-1=\cot ^{2} x$$

Short Answer

Expert verified
The left-hand side of the identity simplifies to \( cot^2(x) \) which matches the right-hand side of the identity. Therefore, the given identity is verified.

Step by step solution

01

Break Down the Identity

Starting with the left-hand side (LHS) of the identity:\( sec^2(\frac{\pi}{2} - x) - 1 \).We know that,\( sec^2(\frac{\pi}{2} - x) = 1 + tan^2(\frac{\pi}{2} - x) \).Therefore,\( 1 + tan^2(\frac{\pi}{2} - x) - 1 = tan^2(\frac{\pi}{2} - x) \).
02

Simplify the Tangent Term

We know that,\( tan(\frac{\pi}{2} - x) = cot(x) \).Therefore,\( tan^2(\frac{\pi}{2} - x) = cot^2(x) \).
03

Compare the Two Sides

Now that the left-hand side is simplified to \( cot^2(x) \), we see that it matches with the right-hand side of the identity. Therefore, it verifies the identity is true.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Functions
Trigonometric functions are essential tools in mathematics, especially when dealing with angles and periodic phenomena.
They describe relationships within a right triangle, with functions like sine (\( \sin \theta \)) and cosine (\( \cos \theta \)) often being the first ones introduced.
Beyond simple right triangle ratios, trigonometric functions are also crucial in calculus and physics.
  • Sine and Cosine: Fundamental functions that relate the opposite or adjacent side, respectively, to the hypotenuse in a right triangle.
  • Tangent: The ratio of sine to cosine, or the opposite side over the adjacent side.
These functions repeat values in a cyclic manner, known as periodicity.
This feature is useful in solving equations and verifying identities.
Functions also transform through shifting, reflecting, and stretching.
Mastering these nuances aids in deeper understanding of mathematical concepts.
Secant Function
The secant function, denoted as sec(\(\theta\)), is the reciprocal of the cosine function: \( \sec \theta = \frac{1}{\cos \theta} \).
This relationship shows that secant is undefined whenever cosine is zero.
In the unit circle framework, the secant represents the horizontal distance from the origin when the terminal side of an angle in standard position intersects the circle.
  • Relation to Tangent: Its Pythagorean identity shows a connection with tangent, given by \( \sec^2 \theta = 1 + \tan^2 \theta \).
  • Angle Transformations: Transforming angles like converting \(\theta\) to \(\frac{\pi}{2} - \theta\) involves using complementary angle identities, often leading to the expression of secant in terms of other functions.
These properties simplify complex expressions and solve trigonometric equations, making the secant function particularly valuable in mathematical proofs.
Cotangent Function
The cotangent function, abbreviated as cot(\( \theta \)), is the reciprocal of the tangent function:\( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \).
This means cotangent is defined whenever sine is non-zero.
The cotangent function can be visually understood using the unit circle where it represents the ratio of the x-coordinate to the y-coordinate when a line from the origin intercepts the circle.
  • Relation to Other Functions: Like tangent, cotangent is periodic with an identity \( \cot^2 \theta + 1 = \csc^2 \theta \).
  • Use in Identities: In identity verification, cotangent functions often appear when evaluating functions at complementary angles, such as \( tan(\frac{\pi}{2} - x) = cot(x) \).
Understanding how cotangent interplays with other trigonometric functions is key in simplifying expressions and verifying trigonometric identities.

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Most popular questions from this chapter

Determine whether the statement is true or false. Justify your answer. Because the sine function is an odd function, for a negative number \(u, \sin 2 u=-2 \sin u \cos u\).

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Verify the identity. $$\frac{\sin x \pm \sin y}{\cos x+\cos y}=\tan \frac{x \pm y}{2}$$

Find all solutions of the equation in the interval \([0,2 \pi)\). $$\sin (x+\pi)-\sin x+1=0$$

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