Question

If sec theta + tan theta = 4 , find sin theta and cos theta

Answer 1

SecØ +TanØ=4
SecØ= 4-TanØ
squaring both sides
Sec²Ø= 16 + Tan²Ø- 8 TanØ
1+Tan²Ø=16 + Tan²Ø- 8 TanØ
we get 8Tan Ø= 15
TanØ=15/8
Tan²Ø= 225/64
Sec²Ø= 1+Tan²Ø
Sec²Ø= 289/64
SecØ =17/8
So CosØ=8/17
Sin²Ø=1-Cos²Ø = 1-64/289
Sin²Ø= 225/289
SinØ= 15/17

Cos Ø= 8/17 and Sin Ø= 15/17
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Answer 2

secθ - tanθ = 4                   -----> (1)

Multiplying and dividing by secθ +tanθ

 = secθ - tanθ × secθ +tanθ/secθ +tanθ

= sec²θ - tan²θ/secθ + tanθ = 4

we know that -->  sec²θ - tan²θ = 1

1/secθ + tanθ = 4

therefore ,

secθ + tanθ = 1/4                  ----> (2)

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Adding equations (1) and (2) :-

secθ - tanθ = 4

secθ + tanθ = 1/4   

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2secθ =  4 + 1/4

2secθ =   17/4

secθ =   17/4*2

         = 17/8

we know that 

cosθ =   1/secθ

         = 1/(17/8)

         = 8/17