Question

5. Prove: Tan beta sin beta + cos beta = sec beta

6. Prove: ( tan y cos^2 y + sin^2 y)/ (sin y) = cos y+sin y

7. Prove (1+tan theta)/(1-tan theta) = (sec^2+2tan theta) /(1-tan^2 theta)

Answer 1

I will use x instead of beta for easier typing

tanxsinx + cosx = secx

I usually try to express everything in terms of sines and cosines, so

LS = (sinx/cos)sinx + cosx
= sin^2x/cosx + cos^2 x/cosx
= (sin^2 x + cos^2 x)/cosx
= 1/cosx
= secx
= RS

Give the other two a try using the same concept