Question

if sinx +sin2x=1,prove that cos2x+cos4x=1

Answer 1

sinx+sin2x=1
or sinx =1-sin2x
=cos2x
Therefore sinx=cos2x........(1)
Therefore sin2x=(cos2x)^2
or sin2x=cos4x.......(2)
Adding equation (1) and (2) we get,
sinx+sin2x=cos2x+cos4x
or cos2x+cos4x=1 (since sinx+sin2x=1)

Answer 2

Answer:

Step-by-step explanation:

Given

sinx +sin²x = 1

To prove

cos²x +  $cos^4x$ = 1

Recall the formula

sin²x + cos²x = 1

Solution

We have, sinx +sin²x = 1

sinx = 1 - sin²x (∵sin²x + cos²x = 1)

sinx = cos²x ----------------------(1)

sin²x = (cos²x)² = $cos^4x$

sin²x  = $cos^4x$ ------------------(2)

LHS  = cos²x +  $cos^4x$ = sinx + sin²x  (from (1) and (2)

= 1 = RHS (from given condition)

∴cos²x +  $cos^4x$ = 1

Hence proved

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