Question

[Nghiêu Nghiêu]

tính giá trị các biểu thức sau:

a, \(A=\left(\sin a+\cos a\right)^2-2\sin a\cos a-1\)

b, \(B=\left(\sin a-\cos a\right)^2+2\sin a\cos a+1\)

c, \(C=\left(\sin a +\cos a\right)^2+\left(\sin a-\cos a\right)^2+2\)

d, \(D=\sin^2a.\cot^2a+\cos^2a.\tan^2a\)

Answer 1

[Phương An]

~ ~ ~ Áp dụng đẳng thức \(\left(a+b\right)^2+\left(a-b\right)^2=2\left(a^2+b^2\right)\) ~ ~ ~

a)

\(\left(\sin\alpha+\cos\alpha\right)^2-2\sin\alpha\cos\alpha-1\)

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

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

= 0

b)

\(\left(\sin\alpha-\cos\alpha\right)^2+2\sin\alpha\cos\alpha+1\)

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

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

\(=2\left(\sin^2\alpha+\cos^2\alpha\right)\)

= 2

c)

\(\left(\sin\alpha+\cos\alpha\right)^2+\left(\sin\alpha-\cos\alpha\right)^2+2\)

\(=2\left(\sin^2\alpha+\cos^2\alpha\right)+2\)

= 4

d)

\(\sin^2\alpha\cot^2\alpha+\cos^2\alpha\tan^2\alpha\)

\(=\left(\sin\times\dfrac{\cos}{\sin}\right)^2+\left(\cos\times\dfrac{\sin}{\cos}\right)^2\)

= 1