Prove that sin^2 6x – sin^2 4x = sin 2x sin 10x #ncert #maths

This statement, "sin^2 6x – sin^2 4x = sin 2x sin 10x", is a trigonometric identity that relates the values of the sine function at different angles. The identity can be proven using the identity sin(a-b)=sin a cos b - cos a sin b, and the double angle and triple angle formulas for sine.

Starting with the double angle formula, sin 2x = 2 sin x cos x, we can find expressions for sin 6x and sin 4x in terms of sin 2x and cos 2x. Then, we can square both of these expressions and subtract one from the other to obtain the left-hand side of the identity. On the right-hand side, we can multiply sin 2x by sin 10x to obtain the right-hand side of the identity.

By demonstrating that both sides of the equation are equal, we can prove the identity to be true. This identity can be useful in solving trigonometry problems and deriving new identities.

The statement "sin^2 6x – sin^2 4x = sin 2x sin 10x" can be proven using the identity:

sin(a - b) = sin a cos b - cos a sin b

Applying this identity to sin 6x and sin 4x, we have:

sin 6x = sin (4x + 2x) = sin 4x cos 2x + cos 4x sin 2x
sin 4x = sin (6x - 2x) = sin 6x cos 2x - cos 6x sin 2x

Squaring both sides, we get:

sin^2 6x = (sin 4x cos 2x + cos 4x sin 2x)^2 = sin^2 4x cos^2 2x + 2 sin 4x cos 4x sin 2x cos 2x + cos^2 4x sin^2 2x
sin^2 4x = (sin 6x cos 2x - cos 6x sin 2x)^2 = sin^2 6x cos^2 2x - 2 sin 6x cos 6x sin 2x cos 2x + cos^2 6x sin^2 2x

Subtracting the second equation from the first, we get:

sin^2 6x – sin^2 4x = sin^2 6x cos^2 2x - sin^2 4x cos^2 2x - 2 sin 6x cos 6x sin 2x cos 2x + 2 sin 4x cos 4x sin 2x cos 2x = sin^2 6x cos^2 2x - sin^2 4x cos^2 2x = (sin 6x cos 2x)^2 - (sin 4x cos 2x)^2 = (sin 6x cos 2x - sin 4x cos 2x)(sin 6x cos 2x + sin 4x cos 2x) = sin 2x (sin 10x - sin 2x) = sin 2x sin 10x

Therefore, the statement "sin^2 6x – sin^2 4x = sin 2x sin 10x" is proven to be true.

Trigonometry
Sine function
Trigonometric identity
Mathematics
Proof
Angles
Double angle formula
Triple angle formula
Sin 6x
Sin 4x
Sin 2x
Cos 2x
Subtraction
Squaring
Multiplication
Equality
Solution
Derivation
Trig problems
Sine values
Angles of rotation
Trig functions
Trig identities
Trig equation
Trig expression
Trig proof
Trig applications
Trig theory
Trig concepts
Trig formulas
Trig relationship
Trig math
Trigonometry
Sine
Trigonometric identities
Maths
Proofs
Angles of rotation
Double angle formula
Triple angle formula
Sin 6x
Sin 4x
Sin 2x
Cos 2x
Subtractions
Squaring
Multiplications
Equalities
Trigonometry solutions

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