Verify the identity sin x cos x 12 2sin x 1cos x 1Solut

Verify the identity. (sin x + cos x + 1)^2 = 2(sin x + 1)(cos x + 1)

Solution

To verify : (sin x + cos x + 1)² = 2 (sin x + 1)(cos x + 1)

Let us take the left hand side of the equation and simplify it.

(sin x + cos x + 1)² =[ sin x + (cos x + 1)²]

= sin² x + 2 sin x (cos x + 1) + (cos x + 1)² [Applying the identity of (a + b)² = a² + 2ab + b²]

= sin² x + 2 sin x (cos x + 1) + cos²x + 2cos x + 1 [Again Applying the identity of (a + b)²]

= sin² x + 2 sin x cos x + 2 sin x + cos²x + 2cos x + 1 [Solving the bracket]

= sin² x + cos²x + 2 sin x cos x + 2 sin x + 2cos x + 1 [Bringing sin² x and cos² x together]

= 1 + 2 sin x cos x + 2 sin x  + 2cos x + 1 [As sin² x + cos²x = 1 ]

= 1 + 1 + 2 sin x + 2 cos x + 2 sin x cos x [Re-arranging the terms]

= 2 + 2 sin x + 2 cos x + 2 sin x cos x

= 2 ( 1 + sin x) + 2 cos x ( 1 + sin x)

= ( 2 + 2 cos x)(1 + sin x)

= 2 (1 + cos x) (1 + sinx )

= 2 (sinx + 1)( cos x + 1) which is equal to the right hand side of the equation.

Hence verified.