How to verify csc 2 ⁡ x cot 2 ⁡ x + csc 2 ⁡ x = csc 4 ⁡ x ?

Question

How to verify       csc    2    ⁡  x      cot    2    ⁡  x  +      csc    2    ⁡  x  =      csc    4    ⁡  x ?

rambandi69v4 · Accepted Answer

Simple trigonometric identities are ones that we are familiar with.

\sin^{2} x + \cos^{2} x = 1 1 + \tan^{2} x = \sec^{2} x 1 + \cot^{2} x = \csc x

To prove

(\csc^{2} x \cot^{2} x + \csc^{2} x) = \csc^{4} x

Taking common term

\csc^{2} x

on L H S out,

\Rightarrow \csc^{2} x (1 + \cot^{2} x)

But

\csc^{2} x = 1 + \cot^{2} x

(Trigonometric identity.)
Then

\Rightarrow \csc^{2} x \cdot \csc^{2} x = \csc^{4} x = R H S

haitwii521 · Accepted Answer

Considering that,                     csc        2            x                      cot        2            x        +                  csc        2            x        =                  csc        4            x            L    H    S    ⇒          1                        sin          2                x              ⋅                            cos          2                x                              sin          2                x              +          1                        sin          2                x                                          cos          2                x                              sin          4                x              +                            sin          2                x                              sin          4                x                                                      cos            2                    x                +                              sin            2                    x                                      sin          4                x                        1                        sin          4                x              =                  csc        4            x

How to verify csc^2xcot^2x+csc^2x=csc^4x ?

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