Question

18) If sin ^4 + sin^2= 1, then prove that
tan^4 - tan² = 1​

Answer 1

⠀⠀ || ✪ ϙᴜᴇsᴛɪᴏɴ ✪ ||

If sin⁴ θ + sin² θ = 1, then prove that

tan⁴ θ - tan² θ = 1 .

⠀⠀|| ☆.ANSWER.☆ ||

Given :-

• sin⁴ θ + sin² θ = 1, ..........(1)

To prove :-

• tan⁴ θ - tan² θ = 1 . ........(2)

|| ☆.Explanation.☆ ||

Take L.H.S. of equ(1),

= tan⁴ θ - tan² θ

Keep,

• tan θ = sin θ/cos θ

= sin⁴ θ/cos⁴ θ - sin² θ /cos² θ

keep by equ(1),

• sin⁴ θ = 1 - sin² θ

= ( 1- sin² θ )/cos⁴ θ - sin² θ /cos² θ

Now, keep

• (1- sin² θ ) = cos² θ

= ( cos² θ )/cos⁴ θ - sin² θ /cos² θ

= 1/cos² θ - sin² θ /cos² θ

= ( 1- sin² θ )/cos² θ

= cos² θ / cos² θ

= 1

= R.H.S.

That's proved.

★Important Formula

➠ tan x = sin x / cos x

➠ cot x = cos x / sin x

➠ sin x = 1/cosec x

➠ cos x = 1/sec x

➠sin x . cosec x = 1

➠ tan x . cot x = 1

➠ sin² x + cos² x = 1

➠ cosec² x - tan² x = 1

➠ sec² x - tan² x = 1

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