Question

show that 1/1+sin theta +. 1/1-sin theta =2sec²theta​

Answer 1

Question : -

Prove that (1)/(1+sin x) + (1)/(1-sin x) = 2sec² x

ANSWER

Given : -

(1)/(1+sin x) + (1)/(1-sin x) = 2sec² x

Required to prove : -

• LHS = RHS

Proof : -

(1)/(1+sin x) + (1)/(1-sin x) = 2sec² x

Consider the LHS part;

(1)/(1+sin x) + (1)/(1-sin x)

Multiply numerator & denominator of the 1st part of LHS with 1-sin x

(1)/(1+sin x) x (1-sin x)/(1-sin x) + (1)/(1-sin x)

(1-sin x)/([1]²-[sin x]²) + (1)/(1-sin x)

(1-sin x)/(1²-sin² x) + (1)/(1-sin x)

(1-sin x)/(1-sin² x) + (1)/(1-sin x)

Multiply numerator & denominator of the 2nd part of LHS with 1+sin x

(1-sin x)/(1-sin² x) + (1)/(1-sin x) x (1+sin x)/(1+sin x)

(1-sin x)/(1-sin² x) + (1+sin x)/([1]²-[sin x]²)

(1-sin x)/(1-sin² x) + (1+sin x)/(1²-sin² x)

(1-sin x)/(1-sin² x) + (1+sin x)/(1-sin² x)

Since, denominator let's take the LCM

(1-sin x + 1+sin x)/(1-sin² x)

(1+1)/(1-sin² x)

(2)/(1-sin² x)

From the identity;

• sin² x + cos² x = 1
• => cos² x = 1-sin² x

(2)/(cos² x)

This is can be written as;

2 x (1)/(cos² x)

Since,

• 1/cos x = sec x

2 x sec² x

2sec² x

Consider the RHS part;

2sec² x

LHS = RHS

Hence Proved ✓