Question

Cho góc \(\alpha\) \((0< \alpha< 90\)\()\) . Tính giá trị biểu thức: B\(=(1-\sin^4\alpha-\cos^4\alpha)(\tan^2\alpha+\cot^2\alpha+2)\).

Answer 1

Lời giải:

Nhớ rằng \(\cos ^2a+\sin ^2a=1\). Ta có:

\(B=(1-\sin ^4a-\cos ^4a)(\tan ^2a+\cot ^2a+2)\)

\(=[1+2\sin ^2a\cos ^2a-(\sin^4a+\cos ^4a+2\sin ^2a\cos ^2a)](\frac{\sin ^2a}{\cos ^2a}+\frac{\cos ^2a}{\sin ^2a}+2)\)

\(=[1+2\sin ^2a\cos ^2a-(\sin ^2a+\cos ^2a)^2].\frac{\sin ^4a+\cos ^4a+2\sin ^2a\cos ^2a}{\cos ^2a\sin ^2a}\)

\(=[1+2\sin ^2a\cos ^2a-1^2].\frac{(\sin ^2a+\cos ^2a)^2}{\cos ^2a\sin ^a}\)

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