Math Questions for points?

Question

2.Prove the identity cos^2x-sin^2x / cos^2x+sinxcosx =cotx-1/cotx

3.Prove that (sinx+cosx) (tan^2x+1/tanx)=1/cosx +1/sinx

Thanks sooo much and god bless

Answer 1

2.

To  prove the identity Presuming cos²x - sin²x / sinxcosx = cotx - 1 / cotx

solve the identity from left hand side

cos²x - sin²x / sinxcosx

Recall : Trigonometry identies cos²x - sin²x = cos2x

cos²x - sin²x / sinxcosx = 2cos2x / 2sinxcosx

Recall : Trigonometry identities 2sinxcosx = sin2x

                                              = 2cos2x / sin2x

Recall : Trigonometry formulas cotA = cosA / sinA

                                              = 2cot2x

Recall : Trigonometry identities cot2x = (cot²x - 1) / 2cotx

                                              = 2( cot²x - 1) / 2cotx

                                             = (cot²x - 1) / cotx

                                             = (cot²x / cotx) - (1 / cotx)

                                             = cotx - (1 / cotx).

Answer 2

3.

Solve the identity from left hand side

(sinx + cosx)((tan²x + 1) / tanx) = (sinx + cosx)(((sin²x / cos²x) + 1) / tanx)

                                                         = (sinx + cosx)[(sin²x + cos²x ) / cos²x × tanx]

Recall : Trigonometry identities sin²x + cos²x = 1

                                                        =  (sinx + cosx)[(1) / cos²x × tanx]

Recall : Trigonometry formulas tanx = sinx / cosx

                                                        = (sinx + cosx)[1 / (cos²x × (sinx / cosx))]

                                                       = (sinx + cosx)( 1 / cosx × sinx)

                                                       = sinx / (cosx × sinx) + cosx / (cosx × sinx)

                                                       = (1 / cosx ) + (1 / sinx) .