Welcome :: Homework Help and Answers :: Mathskey.com

Recent Visits

  
Welcome to Mathskey.com Question & Answers Community. Ask any math/science homework question and receive answers from other members of the community.

13,435 questions

17,804 answers

1,438 comments

120,935 users

Trig Proofs Please Help?

0 votes
Hello,

There are some questions that I am not sure how to do:

1) (sinθ) / (1 + sinθ) = tanθ(secθ - tanθ)

2) cosec^2θ(tan^2θ - sin^2θ) = tan^2θ

3) (secθ) / (tanθ + cotθ) = sinθ

4) (cosθ) / (sinθ + 1) + (sinθ + 1) / (cosθ) = 2secθ

Thank you in advance.
asked Aug 12, 2014 in TRIGONOMETRY by anonymous

2 Answers

0 votes

1) (sinθ) / (1 + sinθ) = tanθ(secθ - tanθ)

Left hand side identity (sinθ) / (1 + sinθ)

Multiply numarator and denominator by 1 - sinθ

= (sinθ) (1 - sinθ) / (1 + sinθ)(1 - sinθ)

= (sinθ) (1 - sinθ) / (1 - sin²θ)

= (sinθ) (1 - sinθ) / (cos²θ)

= (sinθ/cos θ) ((1 - sinθ)/cosθ)

= (tanθ) ((1/cosθ) - ( sinθ/cosθ))

= (tanθ) (secθ - tanθ)

= Right hand side identity.

----------------------------

2) cosec²θ(tan²θ - sin²θ) = tan²θ

Left hand side identity cosec²θ(tan²θ - sin²θ)

= cosec²θtan²θ - cosec²θsin²θ

= (1/sin²θ) (sin²θ/cos²θ) - (1/sin²θ)(sin²θ)

= (1/cos²θ) - 1

= sec²θ - 1

= tan²θ

= Right hand side identity.

answered Aug 12, 2014 by bradely Mentor
0 votes

3) (secθ) / (tanθ + cotθ) = sinθ

Left hand side identity (secθ) / (tanθ + cotθ)

= (1 / cosθ) / [(sinθ / cosθ)+(cosθ / sinθ)]

=(1 / cosθ) / [(sinθsinθ+cosθcosθ) / (sinθcosθ)]

=(1 / cosθ)(sinθcosθ) / (sin²θ + cos²θ)

=sinθ / 1

= sinθ

= Right hand side identity.

----------------------------

4) (cosθ) / (sinθ + 1) + (sinθ + 1) / (cosθ) = 2secθ

Left hand side identity (cosθ) / (sinθ + 1) + (sinθ + 1) / (cosθ)

= [(cosθ)(cosθ)+(sinθ + 1)(sinθ + 1)]/(cosθ)(sinθ + 1)

=[(cos²θ) +(1+sinθ)²]/(cosθ)(sinθ + 1)

=[cos²θ +1+sin²θ+2sinθ]/(cosθ)(sinθ + 1)

=[cos²θ+sin²θ +1+2sinθ]/(cosθ)(sinθ + 1)

=[1 +1+2sinθ]/(cosθ)(sinθ + 1)

=[2+2sinθ]/(cosθ)(sinθ + 1)

=2[1+sinθ]/(cosθ)(sinθ + 1)

=2/cosθ

=2secθ

= Right hand side identity

answered Aug 13, 2014 by bradely Mentor

Related questions

asked Oct 15, 2014 in TRIGONOMETRY by anonymous
asked Aug 22, 2014 in TRIGONOMETRY by anonymous
asked Nov 4, 2014 in TRIGONOMETRY by anonymous
asked Jul 16, 2014 in TRIGONOMETRY by anonymous
...