Question

solve sinx+cosx=1 and solve sec2x=1-tan2x

Answer 1

Heya!!

1) Sin x + Cos x = 1

put sin x = z

z + √ ( 1 - z² ) = 1

becoz Cos x = √ ( 1 - z² )

=>

( z - 1 ) = -√ ( 1 - z² )

SQUARING BOTH SIDES

=>

( z - 1 )² = ( 1 - z² )

=>

z² - 2z +1 = 1 - z²

=>

2z² - 2z = 0

=>

z( z - 1 ) =0

=>

z = 0 or z = 1

=>

Sin x = 0 OR Sin x = 1

Sin x = Sin 0 OR Sin x = Sin 90

x = 0 OR x = 90°

2) Sec 2x = 1 - Tan 2x

=>

1 + Sin 2x = Cos 2x

=>

√{ ( 1 - Cos 2x ) /2 } = Cos 2x

Becoz Cos 2x = 1- 2 Sin² x

=>

Put Cos 2x = z

=>

√ {( 1 - z² ) / 2 } = z

SQUARING BOTH SIDES

=>

( 1 - z² ) = 2z²

=>

-3z² = -1

z = 1/√3 OR z = -1/√3

=>

Cos 2x = 1/√3 OR Cos 2x = -1/√3

2x = Cos-¹ ( 1/√3 ) OR. 2x = Cos-¹ ( -1/√3