Question

simplify ((1+sin u)/cos u )+(cos u)/1+sin u. I solved up to (1+2sin u + 1)/ cos u (1+sin U) what am I missing?

Answer 1

(1 + sinu)/cosu + cosu/(1 + sinu) ← assuming this is what you really mean.

= (1 + sinu)²/[cosu(1 + sinu)] + (cosu)²/[cosu(1 + sinu)]

= (1 + 2sin u + sin²u + cos²u)/[cosu(1 + sinu)]

= (2 + 2sinu)/[cosu(1 + sinu)] (Since sin²u + cos²u = 1)

= 2(1 + sinu)/[cosu(1 + sinu)]

= 2/cosu

= 2secu

Answer 2

$$\frac{1+\sin u}{\cos u}+\frac{\cos u}{1+\sin u}$$First, let's identity a common denominator to reach -- how about \((1+\sin u)\cos u\)? So we write:$$\frac{(1+\sin u)(1+\sin u)}{(1+\sin u)\cos u}+\frac{\cos u\cos u}{(1+\sin u)\cos u}=\frac{(1+\sin u)^2+\cos^2 u}{(1+\sin u)\cos u}$$Simplify our numerator by expanding:$$(1+\sin u)^2+\cos^2u=1+2\sin u+\sin^2u+\cos^2u$$by the Pythagorean theorem, we know \(\cos^2 u+\sin^2 u=1\) so we may simplify our numerator further: $$1+2\sin u+\sin^2u+\cos^2u=1+2\sin u+1=2+2\sin u$$Now let's look at it all together again:$$\frac{2+2\sin u}{(1+\sin u)\cos u}$$You should recognize our numerator factors out to \(2(1+\sin u)\), which means we have a common factor between our numerator and denominator, namely \(1+\sin u\):$$\frac{2(1+\sin u)}{(1+\sin u)\cos u}=\frac2{\cos u}$$We know that \(\frac1{\cos u}=\sec u\) so we finish our reduction by rewriting it using \(\sec u\):$$\frac2{\cos u}=2\sec u$$

Answer 3

thank you both of you. I really appreciate the step by step explanation.

Answer 4

welcome

Answer 5

@Madiyel give out the medal, not just say thank you