Show that: (cot^2 x+1)[sin^2(-x)]+cos((π)/(2)-x) sin(-x)=cos^2 x

Name: Problem 19, Chapter 32: Calculus With Trigonometry and Analytic Geometry
Uploaded: 2021-12-06T04:12:45-08:00
Duration: 1 min 33 s
Channel: Sriparna Bhattacharjee
Description: Show that: (cot^2 x+1)[sin^2(-x)]+cos((π)/(2)-x) sin(-x)=cos^2 x

step 1 Check the question from the text book. To prove cause of tan inverse, sine, cot inverse x is equ

Show that: (cot^2 x+1)[sin^2(-x)]+cos((π)/(2)-x)...

Key Concepts

Algebraic Simplification Using Trigonometric Identities

This concept involves combining trigonometric identities with algebraic manipulation to simplify complex trigonometric expressions. It requires substitution based on identities like the even-odd properties, complementary angles, and Pythagorean relationships, followed by algebraic operations such as factorization and cancellation of terms to prove the equality of expressions.

Pythagorean Identities

Pythagorean identities, such as sin²x + cos²x = 1 and cot²x + 1 = csc²x, are fundamental relationships in trigonometry that relate the squares of sine and cosine, or cotangent and cosecant. These identities are pivotal in transforming and simplifying trigonometric expressions, as they allow the substitution of one form for another to reveal underlying equalities.

Complementary Angle Identities

Complementary angle identities relate the trigonometric functions of an angle to those of its complement. A key example is cos(?/2 – x) = sin x, which is derived from the definitions of sine and cosine on the unit circle. This concept simplifies expressions by replacing cosine of a complementary angle with sine, thereby unifying trigonometric terms.

Even-Odd Trigonometric Identities

These identities describe how trigonometric functions behave under a sign change of the angle. Specifically, sine is an odd function, meaning sin(–x) = –sin x, while cosine is an even function, meaning cos(–x) = cos x. This property is essential in the simplification process of equations involving negative angles, allowing expressions like sin²(–x) to be rewritten as sin²x.

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