\(G=cot^2x-sin^2x.cot^2x+1-cot^2x=1-sin^2x.cot^2x\)

\(=1-sin^2x.\dfrac{cos^2x}{sin^2x}=1-cos^2x=sin^2x\)

2.

\(tana+cota=2\Rightarrow\left(tana+cota\right)^2=4\)

\(\Rightarrow tan^2a+cot^2a+2tana.cota=4\)

\(\Rightarrow tan^2a+cot^2a+2=4\)

\(\Rightarrow tan^2a+cot^2a=2\)

\(A=\sqrt{sin^2x-sin^2x.\frac{cos\text{ }x}{sin\text{ }x}+cos^2x-cos^2x.\frac{sin\text{ }x}{cos\text{ }x}}\)

\(A=\sqrt{\left(sin^2x+cos^2x\right)-\left(sin\text{ }x.cos\text{ }x-cos\text{ }x.sin\text{ }x\right)}\)

\(A=\sqrt{1}=1\)

\(A=\sqrt{\sin^2x\left(1-\cot x\right)+\cos^2x\left(1-\tan x\right)}\)

\(A=\sqrt{\sin^2x-\sin^2x\cot x+\cos^2x-\cos^2x\tan x}\)

\(A=\sqrt{1-\sin^2x\frac{\cos x}{\sin x}-\cos^2x\frac{\sin}{\cos}}\)

\(A=\sqrt{1-\sin x\cos x-\sin x\cos x}\)

\(A=\sqrt{\sin^2x-2\sin x\cos x+\cos^2x}\)

\(A=\sqrt{\left(\sin x-\cos x\right)^2}=\left|\sin x-\cos x\right|\)

Bạn kiểm tra lại đề bài câu 1, câu này chỉ có thể rút gọn đến \(2cot^2x+2cotx+1\) nên biểu thức ko hợp lý

Đồng thời kiểm tra luôn đề câu 2, trong cả 2 căn thức đều xuất hiện \(6sin^2x\) rất không hợp lý, chắc chắn phải có 1 cái là \(6cos^2x\)

Câu 1 đề vẫn có vấn đề:

\(=\dfrac{1+cotx}{1-cotx}-\dfrac{2\left(1+cot^2x\right)cot^2x}{\left(tanx-1\right)\left(tan^2x+1\right)cot^2x}=\dfrac{1+cotx}{1-cotx}-\dfrac{2cot^2x}{tanx-1}\)

\(=\dfrac{1+cotx}{1-cotx}-\dfrac{2cot^3x}{1-cotx}=\dfrac{1+cotx-2cot^3x}{1-cotx}\)

\(=\dfrac{\left(1-cotx\right)\left(1+2cotx+2cot^2x\right)}{1-cotx}=1+2cotx+2cot^2x\)

Có thể coi như ko thể rút gọn tiếp

2.

\(\sqrt{\left(1-cos^2x\right)^2+6cos^2x+3cos^4x}+\sqrt{\left(1-sin^2x\right)^2+6sin^2x+3sin^4x}\)

\(=\sqrt{4cos^4x+4cos^2x+1}+\sqrt{4sin^4x+4sin^2x+1}\)

\(=\sqrt{\left(2cos^2x+1\right)^2}+\sqrt{\left(2sin^2x+1\right)^2}\)

\(=2\left(cos^2x+sin^2x\right)+2=4\)

(sin 1 độ + sin 2 độ + ... + sin 89 độ) - (cos 1 độ + cos 2 độ + ... + cos 89 độ)

=(cos 89 độ +... + cos 2 độ +cos 1 độ) - (cos 1 độ + cos 2 độ + ... + cos 89 độ)

=0

a/ \(A=\left(sin\alpha+cos\alpha\right)^2+\left(sin\alpha-cos\alpha\right)^2=2\left(sin^2\alpha+cos^2\alpha\right)=2\)

b/ \(B=\left(1+tan^2\alpha\right)\left(1-sin^2\alpha\right)-\left(1+cotg^2\alpha\right)\left(1-cos^2\alpha\right)\)

\(=\left(1+\frac{sin^2\alpha}{cos^2\alpha}\right)\left(1-sin^2\alpha\right)-\left(1+\frac{cos^2\alpha}{sin^2\alpha}\right)\left(1-cos^2\alpha\right)\)

\(=\frac{1}{cos^2\alpha}.cos^2\alpha-\frac{1}{sin^2\alpha}.sin^2\alpha=1-1=0\)

\(B=\left(1+\dfrac{sin^2a}{cos^2a}\right).cos^2a-\left(1+\dfrac{cos^2a}{sin^2a}\right).sin^2a\)

\(=\dfrac{\left(sin^2a+cos^2a\right)}{cos^2a}.cos^2a-\left(\dfrac{sin^2a+cos^2a}{sin^2a}\right).sin^2a\)

\(=1-1=0\)

$\sin a.\cos a.(\tan a+\cot a)\\=\sin a.\cos a.\tan a+\sin a.\cos a.\cot a\\=\sin a.\cos a.\dfrac{\sin a}{\cos a}+\sin a.\cos a.\dfrac{\cos a}{\sin a}\\=\sin^2 a+\cos^2 a\\=1$

\(sin\left(a\right).cos\left(a\right).\left(tan\left(a\right)+cot\left(a\right)\right)\\ =sin\left(a\right).cos\left(a\right).tan\left(a\right)+sin\left(a\right).cos\left(a\right).cot\left(a\right)\\ =sin\left(a\right).cos\left(a\right).\dfrac{sin\left(a\right)}{cos\left(a\right)}+sin\left(a\right).cos\left(a\right).\dfrac{cos\left(a\right)}{sin\left(a\right)}\\ =sin^2\left(a\right)+cos^2\left(a\right)=1\)

\(\left(1+\frac{\sin^2}{\cos^2}\right)cos^2-\left(1+\frac{cos^2}{sin^2}\right)sin^2.\)

=> \(\frac{cos^2+sin^2}{cos^2}\left(cos^2\right)-\frac{sin^2+cos^2}{sin^2}\left(sin^2\right)\)

=> 1-1 =0

\(=\frac{1}{cos^2a}\cdot cos^2a+\frac{1}{sin^2a}\cdot sin^2a\) 

\(=1+1\) 

\(=2\)

a, \(\left(1-sin^2x\right)cot^2x+1-cot^2x\)

\(=cot^2x-sin^2x.cot^2x+1-cot^2x\)

\(=1-sin^2x.\frac{\text{cos}^2x}{sin^2x}=1-\text{cos}^2x=sin^2x\)

b,\(\left(tanx+cotx\right)^2-\left(tanx-cotx\right)2\)

\(=tan^2x2.tanx.cotx+cot^2x-tan^2x+2tanx.cotx-cot^2x\)

\(=4tanxcotx=4\)

c,\(\left(xsina-y\text{cos}a\right)^2+\left(x\text{cos}a+ysina\right)^2\)

\(=x^2sin^2a=2xysina\text{cos}a+y^2\text{cos}^2a+2xysina\text{cos}a+y^2sin^2a\)

\(=x^2\left(sin^2a+\text{cos}^2a\right)+y^2\left(sin^2a+\text{cos}^2a\right)\)

\(=x^2+y^2\)