Question

M=cosQ+sinQ and N=cosQ-sinQ

Now prove that,
(1-N)÷(M-1)=cosQ÷(1-sinQ)​

Answer 1

Answer:

\begin{gathered}=\bold{\sqrt{\frac{m}{n}}}+\bold{\sqrt{\frac{n}{m}}}\\\\=\bold{\frac{(m+n)}{\sqrt{mn}}}\end{gathered}

=

n

m

+

m

n

=

mn

(m+n)

now, m = cosq - sinq , n = cosq + sinq

( m + n) = (cosq - sinq) + (cosq + sinq )

= 2cosq

mn = (cosq - sin)(cosq + sinq) = cos²q - sin²q [ as you know, a² -b² = (a -b)(a + b)]

= (cos²q - sin²q)/(sin²q + cos²q) [ as we know, sin²x + cos²x = 1 ]

= (1 - tank)/(1 + tan²q) [ after dividing with cos²q both numerator and denominator]

now,

LHS = (m + n)/√mn

= 2cosq/√{(1 - tan²q)/(1 + tan²q)}

= 2cosq/√{(1 - tan²q)/sec²q} [ sec²x = 1 + tan²x]

= 2cosq.secq/√(1 - tan²q)

= 2/√(1 - tan²q) = RHS