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Chapter 1: Problem 69

Prove the following identities. $$\tan ^{2} \theta+1=\sec ^{2} \theta$$

Short Answer

Expert verified
Question: Prove that $$\tan^2{\theta} + 1 = \sec^2{\theta}$$ using trigonometric identities. Answer: The identity is proven by rewriting the tangent and secant functions in terms of sine and cosine functions, then using algebraic manipulation and the Pythagorean trigonometric identity $$\sin^2{\theta}+\cos^2{\theta} = 1.$$

Step by step solution

01

Rewrite the trigonometric functions

Write the tangent function in terms of sine and cosine functions, and the secant function in terms of cosine function. $$ \tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}} \\ \sec{\theta} = \frac{1}{\cos{\theta}} $$
02

Substitute the definitions into the equation

Replace the tan and sec functions in the original equation with their definitions. $$ \left(\frac{\sin ^{2} \theta}{\cos ^{2} \theta}\right)+1=\frac{1}{\cos ^{2} \theta} $$
03

Simplify the equation

Multiply both sides of the equation by $$\cos^{2}{\theta},$$ then simplify. $$ \sin ^{2} \theta + \cos ^{2} \theta = 1 $$
04

Use a trigonometric identity

Use the Pythagorean identity $$\sin^2{\theta}+\cos^2{\theta} = 1.$$ The equation is true, and the original identity is proven.

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Key Concepts

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

Tangent Function
The tangent function, denoted as \( \tan{\theta} \), is a fundamental trigonometric function. It is defined as the ratio of two other trigonometric functions: sine and cosine. Specifically, it is the quotient of the sine of an angle to the cosine of the same angle.

Mathematically, the relationship is expressed as:
  • \( \tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}} \)
The tangent function is crucial in trigonometry, especially in solving right triangles. It gives the slope of the angle in a right-angled triangle. Since it is derived from sine and cosine, understanding tangent helps with mastering trigonometric identities.

When solving problems involving tangent, remember:
  • Tangent is undefined at angles where cosine is zero (i.e., \( 90^\circ, 270^\circ, ... \)).
  • The tangent function is periodic with a period of \( 180^\circ \) (or \( \pi \) radians).
Secant Function
The secant function, denoted as \( \sec{\theta} \), is another essential trigonometric function. It is related to the cosine function as it is its reciprocal. This means secant is defined as the inverse of cosine.

The formula is straightforward:
  • \( \sec{\theta} = \frac{1}{\cos{\theta}} \)
Being the reciprocal of cosine, the secant function is undefined at angles where the cosine value is zero. Understanding secant is vital when working with trigonometric identities and complex calculations.

Some key points to remember about the secant function include:
  • Secant provides values greater than 1 or less than -1, opposite the range of cosine which is between -1 and 1.
  • The secant function, like tangent, is periodic and repeats every \( 360^\circ \) (or \( 2\pi \) radians).
Pythagorean Identity
The Pythagorean Identity is a key relation in trigonometry that supports various proofs and calculations. It stems from the Pythagorean Theorem in right triangles, and it forms the basis of many trigonometric identities.

The standard form of the Pythagorean Identity is:
  • \( \sin^2{\theta} + \cos^2{\theta} = 1 \)
This identity holds for any angle \( \theta \) and shows the inherent balance between sine and cosine for all angles. Utilizing this identity can simplify calculations and prove identities in trigonometry.

Additional forms involve manipulating the primary identity:
  • Derivative Identity: From the basic identity, by dividing by \( \cos^2{\theta} \), another core identity emerges: \( \tan^2{\theta} + 1 = \sec^2{\theta} \).
  • Similarly, dividing by \( \sin^2{\theta} \) results in another form related to cotangent and cosecant functions.

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Most popular questions from this chapter

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