Question

Prove that

(sin a + cosec a) ²+(Cos a + sec a) ²=7 + tan²a+cot²a

Answer 1

Answer:

(sin a + cosec a) ²+(Cos a + sec a) ²=7 + tan²a+cot²a

Taking LHS

(sin a + cosec a) ²+(Cos a + sec a) ²

= sin²a + cosec²a + 2 sin a cosec a + cos²a + sec²a + 2 cos a sec a

= sin²a + cos²a + 2 sin a cosec a + 2 cos a sec a + cosec²a + sec²a

Now, sin²a + cos²a = 1,

= 1 + 2(sin a cosec a + cos a sec a) + cosec²a + sec²a

Now, cosec²a = 1 + cot²a and sec²a = 1 + tan²a

= 1 + 2( sin a × 1/sin a + cos a × 1/cos a) + 1 + cot²a + 1 + tan²a

= 1 + 2( 1+1) + 1 + cot²a +1 + tan²a

= 1 + 2(2) +2 + cot²a + tan²a

= 1 + 4 + 2 + cot²a + tan²a

= 7 + cot²a + tan²a