Statement -1: If `2"sin"2x - "cos" 2x=1, x ne (2n+1) (pi)/(2), n in Z, "then sin" 2x + "cos" 2x = 5`
Statement-2: `"sin"2x + "cos"2x = (1+2"tan" x - "tan"^(2)x)/(1+"tan"^(2)x)`

Options

A. Statement -1 is true, Statement-2 is true, Statement -2 is a correct explanation for Statement-1.
B. Statement -1 is True, Statement-2 is True, Statement -2 is not a correct explanation for Statement -1.
C. Statement-1 is True, Statement-2 is False.
D. Statement -1 is False, Statement-2 is True.

Answer

D

Explanation

We have,
`"sin" 2x = (2"tan"x)/(1+"tan"^(2)x)" and cos" 2x = (1-"tan"^(2)x)/(1+"tan"^(2)x)`
`therefore "sin"2x+ "cos" 2x = (1+2"tan"x - "tan"^(2)x)/(1+"tan"^(2)x)`
So, statement-2 is true.
Let us now consider statement-1.
We have, `2"sin" 2x-"cos" 2x = 1`
`rArr 2"sin" 2x = 2"cos"^(2)x`
`rArr 2"sin" x "cos" x = "cos"^(2) x`
`rArr "tan" x = (1)/(2) " " [because x ne (2n + 1)(pi)/(2) therefore "cos" x ne 0]`
`therefore "sin"2x + "cos"2x = (1+2"tan"x-"tan"^(2)x)/(1+"tan"^(2)x)`
`rArr "sin" 2x + "cos"2x = (2-(1)/(4))/(1+(1)/(4)) = (7)/(5)`
So, statement-1 is not correct.