Question

7. Prove that tan B sin B + cos B = sec B.

11. Prove that tanλ cos^2λ +sin^2λ/sinλ = cos λ + sin λ.

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

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

Answer 1

tan*sin+cos
= sin^2/cos + cos
= (sin^2+cos^2)/cos
= 1/cos
= sec

(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)

the last one needs some parentheses. Too complicated and ambiguous