Free help with homework

Why join Brainly?

  • ask questions about your assignment
  • get answers with explanations
  • find similar questions


The Brainliest Answer!

This Is a Certified Answer

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.

   Let us multiply the denominator and numerator with a rationalizing factor.  So that we can simplify the denominator. 

We know that sec² θ - tan² θ = 1 
             or, (sec θ - tanθ) (sec θ + tan θ) = 1,    as  A² - B² = (A-B) (A+B)

X=\frac{(sec\theta-tan\theta)^2}{(sec\theta+tan\theta)(sec\theta-tan\theta)}\\\\=\frac{sec^2\theta+tan^2\theta-2sec\theta\ tan\theta}{sec^2\theta-tan^2\theta}\\\\=1+tan^2\theta+tan^2\theta-2sec\theta\tan\theta=\\\\=1-2\ sec\theta\ tan\theta+2\ tan^2\theta

similarly we need to use:  Cosec² Ф - Cot² Ф = 1
   Sin² Ф + Cos ²Ф = 1    while rationalizing  fractions.
if you are comfortable with Sin Ф and Cos Ф, then:

\frac{sec\theta-tan\theta}{sec\theta+tan\theta}\\\\=\frac{\frac{1}{cos\theta}-\frac{sin\theta}{cos\theta}}{\frac{1}{cos\theta}+\frac{sin\theta}{cos\theta}}\\\\=\frac{1-sin\theta}{1+sin\theta}\\\\multiply\ with\ (1-sin\theta)\ numerator\ and\ denominator\\\\=\frac{(1-sin\theta)^2}{1^2-sin^2\theta}\\\\=\frac{1+sin^2\theta-2sin\theta}{cos^2\theta}\\\\=\frac{1}{cos^2\theta}+\frac{sin^2\theta}{cos^2\theta}-\frac{2\sin\theta}{cos\theta\ cos\theta}\\\\=sec^2\theta+tan^2\theta-2tan\theta\ sec\theta\\\\.....

2 5 2
how did u get the 2nd step sir???
thank u sir!
The Brain
  • The Brain
  • Helper
Not sure about the answer?
Learn more with Brainly!
Having trouble with your homework?
Get free help!
  • 80% of questions are answered in under 10 minutes
  • Answers come with explanations, so that you can learn
  • Answer quality is ensured by our experts