If sinx+sin^2 x+sin^3 x=1, then cos^6 x-4 cos^4 x+8 cos^2 x is equal to (a) 0 (b) 2 (c) 4 (d) 8

Name: Problem 4, Chapter 10: Comprehensive Mathematics for JEE Advanced
Uploaded: 2022-03-22T06:13:11-08:00
Duration: 3 min 47 s
Channel: Ankit Singh
Description: If sinx+sin^2 x+sin^3 x=1, then cos^6 x-4 cos^4 x+8 cos^2 x is equal to (a) 0 (b) 2 (c) 4 (d) 8

step 1 So basically we need to find the correct option as per the given expression. So if we see this e

If sinx+sin^2 x+sin^3 x=1, then cos^6 x-4 cos^4 x+8 cos^2 x...

Key Concepts

Power-Reduction Techniques

These techniques involve rewriting higher power trigonometric functions, such as cos?x or cos?x, in terms of functions of lower powers or in forms that are easier to manipulate. Power-reduction formulas and algebraic factorization help simplify complex expressions and make it possible to evaluate them without direct computation of high exponents.

Trigonometric Identities

These are fundamental relations between sine, cosine, and other trigonometric functions that allow one to transform and simplify expressions. They include formulas like the angle sum and difference identities, double angle formulas, and product-to-sum identities, all of which are crucial when moving between different trigonometric expressions.

Pythagorean Identity

This identity, sin²x + cos²x = 1, is essential in relating the sine and cosine functions. It is commonly used to substitute one function for another and to transform equations so that they can be solved by reducing the number of distinct trigonometric functions in an expression.

Polynomial Equations in Trigonometric Functions

Many trigonometric equations can be expressed as polynomial equations by treating a trigonometric function as a variable. For instance, an equation involving sin x, sin²x, and sin³x can be seen as a cubic polynomial in sin x, which then can be approached with standard algebraic techniques such as factoring or using the quadratic formula for reduced degrees.

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