Question

যদি $sin\theta+cos\theta=a$ হয়, তবে $sin^4\theta+cos^4\theta$ এর মান কত?

Answer 1

দেওয়া আছে,
$sin\theta+cos\theta=a$

বা, $(sin\theta+cos\theta)^2=a^2$

বা, $sin^2\theta+2\cdot sin\theta \cdot cos\theta+cos^2\theta=a^2$

বা, $(sin^2\theta+cos^2\theta)+2\cdot sin\theta \cdot cos\theta=a^2$

বা, $1+2\cdot sin\theta \cdot cos\theta=a^2$

বা, $2\cdot sin\theta \cdot cos\theta=a^2-1$

বা, $sin\theta \cdot cos\theta=\frac{a^2-1}{2}$

এখন প্রদত্ত রাশি,
$sin^4\theta+cos^4\theta$

$=(sin^2\theta)^2+(cos^2\theta)^2$

$=(sin^2\theta+cos^2\theta)^2-2\cdot sin^2\theta \cdot cos^2\theta$

$=(sin^2\theta+cos^2\theta)^2-2(sin\theta \cdot cos\theta)^2$

$=(1)^2-2\left(\frac{a^2-1}{2}\right)^2$
[মান বসিয়ে]

$=1-2\times\frac{(a^2-1)^2}{4}$

$=1-\frac{(a^2-1)^2}{2}$ [Answer]

Comments

Rules Applied:

• $sin^2\theta+cos^2\theta=1$
• $(a+b)^2=a^2+2ab+b^2$
• $a^2+b^2=(a+b)^2-2ab$