Question

6. Prove that tan λ cos^2 λ + sin^2λ/sin λ
= cos λ + sin λ

10. Prove that 1+tanθ/1-tanθ = sec^2θ+2tanθ/1-tan^2θ

17.Prove that sin^2w-cos^2w/tan w sin w + cos w tan w = cos w-cot w cos w

23. Find a counterexample to shows that the equation sec a – cos a = sin a sec a is not an identity.

Answer 1

PLEASE tell your classmates to use parentheses to clarify the problems!

I think you mean
(tan*cos^2 + sin^2)/sin
= (sin*cos + sin^2)/sin
= sin(cos+sin)/sin
= cos+sin

I think you mean
(1+tan)/(1-tan)
= (1+tan)^2/(1-tan^2)
= (1+2tan+tan^2)/(1-tan^2)
= (sec^2+2tan)/(1-tan^2)

I think you mean
(sin^2 - cos^2)/(tan*sin+cos*tan)
= (sin-cos)(sin+cos)/(tan(sin+cos)
= (sin-cos)/tan
= sin*cot - cos*cot
= cos-cos*cot

always try simple values for counter-examples:
a=0: 1-1 =? 0 -- yes
a=pi: -1+1 =? 0 -- yes
a=pi/4: √2 - 1/√2 =? 1/√2*√2 NO!