How to prove the two difficult trigonometric identities?

[Guest]

Can you prove the following?

[tex]{sec}^{6}(x) - {tan}^{6}(x) = 1 + 3 ~{tan}^{2}(x)~{sec}^{2}(x)[/tex]

[tex]sin^2(x) ~ tan(x) + cos^2(x) ~ cot(x) + 2 ~ sin(x) ~ cos(x) = tan(x) + cot(x)[/tex]


If not, the following free math tutoring video may be helpful:

https://www.youtube.com/watch?v=ik0gZN6MYUM

[Guest]

[tex]\triangle[/tex]ABC -rectangular
BC=a ,AC=b ,AB=c ,[tex]\angle[/tex]ACB=90[tex]^\circ[/tex] ,[tex]\angle[/tex]BAC=x

[tex]sin^{2 }[/tex]x.tgx+[tex]cos^{2 }[/tex]x.cotgx+2sinx.cosx =? tgx+cotgx

([tex]\frac{a}{c}) ^{2 }[/tex].[tex]\frac{a}{b}[/tex]+([tex]\frac{b}{c}) ^{2 }[/tex].[tex]\frac{b}{a}[/tex]+2.[tex]\frac{a}{c}[/tex].[tex]\frac{b}{c}[/tex] =? [tex]\frac{a}{b}[/tex]+[tex]\frac{b}{a}[/tex]

[tex]\frac{ a^{3 } }{b c^{2 } }[/tex]+[tex]\frac{ b^{3 } }{a c^{2 } }[/tex]+[tex]\frac{2ab}{ c^{2 } }[/tex]=? [tex]\frac{ a^{2 }+ b^{2 } }{ab}[/tex]

new denominator ab.[tex]c^{2 }[/tex]

[tex]\frac{ a^{4 }+ b^{4 }+2 a^{2 } b^{2 } }{ab c^{2 } }[/tex]=? [tex]\frac{ a^{2 } c^{2 } + b^{2 } c^{2 } }{ab c^{2 } }[/tex]

([tex]a^{2 }) ^{2 }[/tex]+2[tex]a^{2 }[/tex][tex]b^{2 }[/tex]+([tex]b^{2 } ) ^{2 }[/tex]=? [tex]c^{2 }[/tex]([tex]a^{2 }[/tex]+[tex]b^{2 }[/tex])

([tex]a^{2 }+ b^{2 } )^{2 }[/tex]=? [tex]c^{2 }[/tex].[tex]c^{2 }[/tex]

([tex]c^{2 }) ^{2 }[/tex]=? [tex]c^{4 }[/tex]

[tex]c^{4 }[/tex]=? [tex]c^{4 }[/tex]

[Guest]

[tex]\triangle[/tex]ABC -rectangular
BC=a ,AC=b ,AB=c ,[tex]\angle[/tex]ACB=90[tex]^\circ[/tex] ,[tex]\angle[/tex]BAC=x

[tex]sin^{2 }[/tex]x.tgx+[tex]cos^{2 }[/tex]x.cotgx+2sinx.cosx =? tgx+cotgx

([tex]\frac{a}{c}) ^{2 }[/tex].[tex]\frac{a}{b}[/tex]+([tex]\frac{b}{c}) ^{2 }[/tex].[tex]\frac{b}{a}[/tex]+2.[tex]\frac{a}{c}[/tex].[tex]\frac{b}{c}[/tex] =? [tex]\frac{a}{b}[/tex]+[tex]\frac{b}{a}[/tex]

[tex]\frac{ a^{3 } }{b c^{2 } }[/tex]+[tex]\frac{ b^{3 } }{a c^{2 } }[/tex]+[tex]\frac{2ab}{ c^{2 } }[/tex]=? [tex]\frac{ a^{2 }+ b^{2 } }{ab}[/tex]

new denominator ab.[tex]c^{2 }[/tex]

[tex]\frac{ a^{4 }+ b^{4 }+2 a^{2 } b^{2 } }{ab c^{2 } }[/tex]=? [tex]\frac{ a^{2 } c^{2 } + b^{2 } c^{2 } }{ab c^{2 } }[/tex]

([tex]a^{2 }) ^{2 }[/tex]+2[tex]a^{2 }[/tex][tex]b^{2 }[/tex]+([tex]b^{2 } ) ^{2 }[/tex]=? [tex]c^{2 }[/tex]([tex]a^{2 }[/tex]+[tex]b^{2 }[/tex])

([tex]a^{2 }+ b^{2 } )^{2 }[/tex]=? [tex]c^{2 }[/tex].[tex]c^{2 }[/tex]

([tex]c^{2 }) ^{2 }[/tex]=? [tex]c^{4 }[/tex]

[tex]c^{4 }[/tex]=? [tex]c^{4 }[/tex]

Thanks for sharing the solution but I am a beginner and need to give a university level test. Actually, I need to practice these questions https://www.youtube.com/watch?v=2kHUfdUdYkg but it is hard to apply the same method for different questions like this. Can you refer to any book or online source where I can learn and solve the basic level of questions? But they should be in English instead of Indonesian languages.

I am good at finding angles and hypotenuse questions but now, I want to enhance my learning level.