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Chapter 9: Problem 15

Show that (a) \(\tan ^{2} \theta+1=\sec ^{2} \theta\) (b) \(1+\cot ^{2} \theta=\operatorname{cosec}^{2} \theta\)

Short Answer

Expert verified
Question: Prove the following trigonometric identities: a) \(tan^2\theta + 1 = sec^2\theta\) b) \(1 + cot^2\theta = cosec^2\theta\) Answer: a) We proved that the equation \(\tan^2\theta+1=\sec^2\theta\) is true by rewriting the tangent and secant functions in terms of sine and cosine functions, and then manipulating the Pythagorean identity. b) Similarly, we proved that the equation \(1+\cot^2\theta=\operatorname{cosec}^2\theta\) is true by rewriting the cotangent and cosecant functions in terms of sine and cosine functions, and then manipulating the Pythagorean identity.

Step by step solution

01

(Part a: Rewrite the tan function)

Begin by rewriting the \(\tan^2\theta\) function in terms of sine and cosine functions. Recall that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), so \(\tan^2\theta = \frac{\sin^2\theta}{\cos^2\theta}\).
02

(Part a: Rewrite the sec function)

Similarly, rewrite the \(\sec^2\theta\) function in terms of cosine functions. Recall that \(\sec \theta = \frac{1}{\cos \theta}\), so \(\sec^2\theta = \frac{1}{\cos^2\theta}\).
03

(Part a: Add the Pythagorean identity)

Now, include the Pythagorean identity, which states that \(\sin^2 \theta + \cos^2 \theta = 1\). We want to manipulate this identity to make the left-hand side of the equation look like our starting point: \(\tan^2\theta +1\).
04

(Part a: Divide both sides by \(\cos^2 \theta\))

Divide both sides of the Pythagorean identity by \(\cos^2\theta\) to get \(\frac{\sin^2\theta}{\cos^2\theta}+\frac{\cos^2\theta}{\cos^2\theta}=\frac{1}{\cos^2\theta}\).
05

(Part a: Simplify and compare)

Simplify the terms to get \(\frac{\sin^2\theta}{\cos^2\theta}+1=\frac{1}{\cos^2\theta}\). Notice that this is equivalent to the statement \(\tan^2\theta+1=\sec^2\theta\), proving that the equation holds true.
06

(Part b: Rewrite the cot function)

Begin by rewriting the \(\cot^2\theta\) function in terms of sine and cosine functions. Recall that \(\cot \theta = \frac{\cos \theta}{\sin \theta}\), so \(\cot^2\theta = \frac{\cos^2\theta}{\sin^2\theta}\).
07

(Part b: Rewrite the cosec function)

Similarly, rewrite the \(\operatorname{cosec}^2\theta\) function in terms of sine functions. Recall that \(\operatorname{cosec} \theta = \frac{1}{\sin \theta}\), so \(\operatorname{cosec}^2\theta = \frac{1}{\sin^2\theta}\).
08

(Part b: Add the Pythagorean identity)

Now, include the Pythagorean identity again, which states that \(\sin^2 \theta + \cos^2 \theta = 1\). We want to manipulate this identity to make the left-hand side of the equation look like our starting point: \(1+\cot^2\theta\).
09

(Part b: Divide both sides by \(\sin^2 \theta\))

Divide both sides of the Pythagorean identity by \(\sin^2\theta\) to get \(\frac{\sin^2\theta}{\sin^2\theta}+\frac{\cos^2\theta}{\sin^2\theta}=\frac{1}{\sin^2\theta}\).
10

(Part b: Simplify and compare)

Simplify the terms to get \(1+\frac{\cos^2\theta}{\sin^2\theta}=\frac{1}{\sin^2\theta}\). Notice that this is equivalent to the statement \(1+\cot^2\theta=\operatorname{cosec}^2\theta\), proving that the equation holds true.

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