Prove the identity: (cos θ + sin θ)/(cosθ-sinθ) ≡ sec 2θ + tan 2θ

First rewrite right hand side in terms of sinθ and cosθ, because those are the terms we'll be dealing with on the left hand side: sec2θ+tan2θ = 1/cos2θ + sin2θ/cos2θ, so RHS = (1+sin2θ)/cos2θNow look at the LHS side terms. We probably want to get rid of the cosθ-sinθ on the bottom line to try and get the LHS to look like the RHS. Try multiplying by (cosθ+sinθ) on top and bottom: gives (cos2θ+sin2θ+ 2cosθsinθ)/(cos2θ-sin2θ)Now apply double angle formulas: cos2θ+sin2θ=1 sin2θ= 2cosθsinθ cos2θ-sin2θ=cos2θsubstituting in with these formulas leaves: (1+sin2θ)/cos2θwhich, as we worked out at the start, is equal to sec2θ+tan2θ!

Answered by Margot M. Maths tutor

7164 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

An open-topped fish tank is to be made for an aquarium. It will have a square base, rectangular sides, and a volume of 60 m3. The base materials cost £15 per m2 and the sides £8 per m2. What should the height be to minimise costs?


Solve |3x+1| = 1


A particle A of mass 0.1kg is moving at a speed of 1.5m/s to the right. It collides with a particle B of mass 0.3kg moving at a speed of 1.1m/s to the right. Calculate change in momentum of particle A if particle B has a speed of 1.4m/s after collision.


Let f(x)=e^x sin(x^2). Find f'(x)


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences