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!

MM
Answered by Margot M. Maths tutor

7221 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

Show that 1+cot^2(x)=cosec^2(x)


AS Maths ->Expresss x^2 + 3x + 2 in the form (x+p)^2 + q... where p and q are rational number


Express the equation cosecθ(3 cos 2θ+7)+11=0 in the form asin^2(θ) + bsin(θ) + c = 0, where a, b and c are constants.


A man travels 360m along a straight road. He walks for the first 120m at 1.5ms-1, runs the next 180m at 4.5ms-1, and then walks the final 60m at 1.5ms-1. A women travels the same route, in the same time. At what time does the man overtake the women?


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