Question

Tan x ( 1+sec2x)(1+sec4x)(1+sec8x) is equal to

Answer 1

Answer:

Tan8x

Step-by-step explanation:

Tanx(1 + Sec2x)(1 + Sec4x)(1 + Sec8x)

= Tanx(1 + Cos2x)(1 + Cos4x)(1 + Cos8x) / Cos2x * Cos4x * Cos8x

Remember 1 + Cos2x = 2Cos²x and extend it to cos4x and cos8x as well.

= Tanx(2Cos²x)(2Cos²2x)(2Cos²4x)/ Cos2x * Cos4x * Cos8x

= 8 Tanx(Cos²xCos2x.Cos4x)/Cos8x.

= 8 SinxCosxCos2x.Cos4x / Cos8x

Remember Sin2x = 2SinxCosx

= 4Sin2xCos2xCos4x/Cos8x

= 2. Sin4xCos4x/Cos8x

= Sin8x/Cos8x

= Tan8x.

Answer 2

Concept

Trigonometry is a branch of mathematics which is based on specific function of angles and their applications.

Find

tanx( 1+sec2x)(1+sec4x)(1+sec8x)

Solution

tan x ( 1+sec2x)(1+sec4x)(1+sec8x)

secx = 1/cosx

sec2x = 1/cos2x

sec4x = 1/cos4x

sec8x = 1/cos8x

tanx(1 + cos2x)(1 + cos4x)(1 + cos8x) / cos2x * cos4x * cos8x

We know that

1 + cos2x = 2cos²x

1 + cos4x = 2cos²2x

1 + cos8x = 2cos²4x

= tanx(2cos²x)(2cos²2x)(2cos²4x)/ cos2x * cos4x * cos8x

= 8tanx(cos²xcos2x.cos4x)/cos8x.

tanx = sinx /cosx

= 8 sinxcosxcos2x.cos4x / cos8x

We also know that

sin2x = 2sinxcosx

sin4x = 2sin2xcos2x

sin8x = 2sin4xcos4x

So,

= 4 (2sinxcosx) cos2x cos4x/cos8x

= 4sin2x cos2x cos4x / cos8x

= 2(2sin2xcos2x)cos4x/cos8x

= 2sin4xcos4x/cos8x

=sin8x/cos8x

= tan8x

tanx( 1+sec2x)(1+sec4x)(1+sec8x) is equal to tan8x

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