7.1 Trigonometric Identities

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7.1.1 Learning Objectives

By the end of this section, you should be able to:

• simplify expressions using fundamental identities;
• convert among trigonometric functions;
• verify identities algebraically;
• choose productive identity-proof strategies; and
• distinguish an identity from an equation.

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7.1.2 Fundamental Identities

The reciprocal, quotient, and Pythagorean identities include

\[ \csc x=\frac1{\sin x},\quad \sec x=\frac1{\cos x},\quad \cot x=\frac1{\tan x}, \]

\[ \tan x=\frac{\sin x}{\cos x},\qquad \cot x=\frac{\cos x}{\sin x}, \]

\[ \sin^2x+\cos^2x=1,\quad 1+\tan^2x=\sec^2x,\quad 1+\cot^2x=\csc^2x. \]

Example 7.1 Simplifying with Identities

Simplify \(\dfrac{1-\cos^2x}{\sin x}\).

Solution

\[ \frac{1-\cos^2x}{\sin x} =\frac{\sin^2x}{\sin x} =\sin x. \]

\[ \boxed{\sin x} \]

Exercise 7.1 Simplify \(\dfrac{\sec^2x-1}{\tan x}\).

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Answer: \(\boxed{\tan x}\)

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7.1.3 Verifying Identities

To verify an identity, work with one side at a time until it matches the other. Useful strategies include rewriting in sine and cosine, factoring, combining fractions, multiplying by a conjugate, and using a Pythagorean identity.

Example 7.2 Verifying an Identity

Verify

\[ \frac{1-\sin x}{\cos x}=\frac{\cos x}{1+\sin x}. \]

Solution

Starting with the left side and multiplying by a conjugate,

\[ \frac{1-\sin x}{\cos x}\cdot\frac{1+\sin x}{1+\sin x} =\frac{1-\sin^2x}{\cos x(1+\sin x)} =\frac{\cos^2x}{\cos x(1+\sin x)} =\frac{\cos x}{1+\sin x}. \]

Thus, the identity is verified wherever both sides are defined.

Exercise 7.2 Verify \(\tan x\cos x=\sin x\).

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Answer: \(\tan x\cos x=(\sin x/\cos x)\cos x=\boxed{\sin x}\).

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7.1.4 Identity-Proof Strategies

Do not cancel terms across addition or subtraction. If both sides are complicated, simplify the more complex side first. Preserve domain restrictions even when factors cancel.

Example 7.3 Combining Fractions

Verify

\[ \frac1{1-\cos x}+\frac1{1+\cos x}=2\csc^2x. \]

Solution

\[ \begin{aligned} \frac1{1-\cos x}+\frac1{1+\cos x} &=\frac{2}{1-\cos^2x}\\ &=\frac2{\sin^2x}\\ &=2\csc^2x. \end{aligned} \]

Exercise 7.3 Simplify \((\sec x-\tan x)(\sec x+\tan x)\).

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Answer: \(\sec^2x-\tan^2x=\boxed{1}\).

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7.1.5 Conceptual Takeaways

• Identities state relationships valid for every common-domain input.
• Fundamental identities allow expressions to change form without changing value.
• Verification transforms one side rather than manipulating the identity as an equation.
• Factoring, common denominators, and conjugates remain important algebraic tools.
• Simplification does not erase original domain restrictions.

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7.1.6 Skills You Should Be Able to Do

• Apply reciprocal, quotient, and Pythagorean identities.
• Rewrite expressions in sine and cosine.
• Verify identities using organized algebra.
• Choose an efficient proof strategy.
• Track excluded inputs.

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7.1.7 Practice Problems with Solutions

1. Simplify \(\sin x\csc x\).

   Show Solution

   Since \(\csc x=1/\sin x\), the product is \(\boxed{1}\) wherever defined.

2. Simplify \(\cos x\tan x\).

   Show Solution

   \[ \cos x\left(\frac{\sin x}{\cos x}\right)=\boxed{\sin x}. \]

3. Simplify \(\dfrac{1-\sin^2x}{\cos x}\).

   Show Solution

   \[ \frac{\cos^2x}{\cos x}=\boxed{\cos x}. \]

4. Verify \((1+\tan^2x)\cos^2x=1\).

   Show Solution

   \[ (1+\tan^2x)\cos^2x=\sec^2x\cos^2x=\boxed{1}. \]

5. Verify \(\dfrac{\sin x}{1+\cos x}=\dfrac{1-\cos x}{\sin x}\).

   Show Solution

   Multiply the left side by \((1-\cos x)/(1-\cos x)\):

   \[ \frac{\sin x(1-\cos x)}{1-\cos^2x} =\frac{\sin x(1-\cos x)}{\sin^2x} =\boxed{\frac{1-\cos x}{\sin x}}. \]

6. Simplify \(\dfrac{\csc x-\sin x}{\cos x}\).

   Show Solution

   \[ \frac{1/\sin x-\sin x}{\cos x} =\frac{1-\sin^2x}{\sin x\cos x} =\frac{\cos^2x}{\sin x\cos x} =\boxed{\cot x}. \]

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