tính giá trị biểu thức: a) A=tg 20 . tg35 . tg45 . tg55 . tg70 b) B=sin^6 $\alpha$ + cos ^6 $\alpha$ +3sin^2 $\alpha$ . cos^2 $\alpha$ c) C=3(sin^4 $\alpha$ + cos^4 $\alpha$ -2(sin^6 $\alpha$ + cos ^6 $\alpha$) d) D=căn(sin^4 $\alpha$ + 4cos^2 $\alpha$) + căn(cos^4 $\alpha$ + 4sin^2 $\alpha$)

Đáp án:

`a)` `A=1`

`b)` `B=1`

`c)` `C=1`

`d)` `D=3`

Giải thích các bước giải:

`a)`

`A=\tan20^{0}.\tan35^{0}.\tan45^{0}.\tan55^{0}.\tan70^{0}`

`A=\cot(90^0-20^0).\tan35^{0}.\tan45^{0}.\cot(90^0-55^0).\tan70^{0}`

`A=\cot70^{0}.\tan35^{0}.\tan45^{0}.\cot35^{0}.\tan70^{0}`

`A=(\tan70^{0}.\cot70^{0}).(\tan35^{0}.\cot35^{0}).\tan45^0`

`A=1.1.1`

`A=1`

Vậy `A=1`

` `

`b)`

`B=\sin^6\alpha+\cos^6\alpha+3\sin^2\alpha.\cos^2\alpha`

`B=\sin^6\alpha+\cos^6\alpha+3\sin^2\alpha.\cos^2\alpha.1`

`B=\sin^6\alpha+\cos^6\alpha+3\sin^2\alpha.\cos^2\alpha.(\sin^2\alpha+\cos^2\alpha)`

`B=(\sin^2\alpha)^3+3\sin^4\alpha.\cos^2\alpha+3\sin^2\alpha.\cos^4\alpha+(\cos^2\alpha)^3`

`B=(\sin^2\alpha+\cos^2\alpha)^3`

`B=1^3`

`B=1`

Vậy `B=1`

` `

`c)`

`C=3(\sin^4\alpha+\cos^4\alpha)-2(\sin^6\alpha+\cos^6\alpha)`

`C=3[(\sin^4\alpha+2\sin^2\alpha.\cos^2\alpha+\cos^4\alpha)-2\sin^2\alpha.\cos^2\alpha]-2[(\sin^2\alpha)^3+(\cos^2\alpha)^3]`

`C=3[(\sin^2\alpha+\cos^2\alpha)^2-2\sin^2\alpha.\cos^2\alpha]-2[(\sin^2\alpha+\cos^2\alpha)(\sin^4-\sin^2\alpha.\cos^2\alpha+\cos^4\alpha)]`

`C=3(1-2\sin^2\alpha.\cos^2\alpha)-2[1.(\sin^4\alpha+\cos^4\alpha-\sin^2\alpha.\cos^2\alpha)]`

`C=3(1-2\sin^2\alpha.\cos^2\alpha)-2(\sin^4\alpha+\cos^4\alpha-\sin^2\alpha.\cos^2\alpha)`

`C=3(1-2\sin^2\alpha.\cos^2\alpha)-2[(\sin^4\alpha+2\sin^2\alpha.\cos^2\alpha+\cos^4\alpha)-2\sin^2\alpha.\cos^2\alpha-\sin^2\alpha.\cos^2\alpha]`

`C=3(1-2\sin^2\alpha.\cos^2\alpha)-2[(\sin^2\alpha+\cos^2\alpha)^2-3\sin^2\alpha.\cos^2\alpha]`

`C=3(1-2\sin^2\alpha.\cos^2\alpha)-2(1-3\sin^2\alpha.\cos^2\alpha)`

`C=3-6\sin^2\alpha.\cos^2\alpha-2+6\sin^2\alpha.\cos^2\alpha`

`C=(3-2)+(6\sin^2\alpha.\cos^2\alpha-6\sin^2\alpha.\cos^2\alpha)`

`C=1+0`

`C=1`

Vậy `C=1`

` `

`d)`

`D=\sqrt{\sin^4\alpha+4\cos^2\alpha}+\sqrt{\cos^4\alpha+4\sin^2\alpha}`

`D=\sqrt{(\sin^2\alpha)^2+4\cos^2\alpha}+\sqrt{(\cos^2\alpha)^2+4\sin^2\alpha}`

`D=\sqrt{(\sin^2\alpha+\cos^2\alpha-\cos^2\alpha)^2+4\cos^2\alpha}+\sqrt{(\sin^2\alpha+\cos^2\alpha-\sin^2\alpha)^2+4\sin^2\alpha}`

`D=\sqrt{(1-\cos^2\alpha)^2+4\cos^2\alpha}+\sqrt{(1-\sin^2\alpha)^2+4\sin^2\alpha}`

`D=\sqrt{1-2\cos^2\alpha+\cos^4\alpha+4\cos^2\alpha}+\sqrt{1-2\sin^2\alpha+\sin^4\alpha+4\sin^2\alpha}`

`D=\sqrt{1+2\cos^2\alpha+\cos^4\alpha}+\sqrt{1+2\sin^2\alpha+\sin^4\alpha}`

`D=\sqrt{(1+\cos^2\alpha)^2}+\sqrt{(1+\sin^2\alpha)^2}`

`D=|1+\cos^2\alpha|+|1+\sin^2\alpha|`

`D=1+\cos^2\alpha+1+\sin^2\alpha` 

`D=2+(\sin^2\alpha+\cos^2\alpha)`

`D=2+1`

`D=3`

Vậy `D=3`