Question

Prove that :(sinx+cosecx)^2+(cosx+secx)^2=7+tan^2x+cot^2x

Answer 1

Hey there !!

Prove that :-

→ ( sin x + cosec x )² + ( cos x + sec x )² = 7 + tan²x + cot²x .

Solution :-

→ Solving  LHS,

= ( sin x + cosec x )² + ( cos x + sec x )² .

= ( sin²x + cosec²x + 2sin x cosec x ) + ( cos²x + sec²x + 2cos x sec x ) .

= ( sin²x + cosec²x + 2 ) + ( cos²x + sec²x + 2 ) .

        [ ∵ sin x cosec x = 1 and cos x cosec x = 1 ] .

= sin²x + cosec²x + 2 + cos²x + sec²x + 2 .

= ( sin²x + cos²x ) + 4 + ( cosec²x + sec²x ) .

= 1 + 4 + ( 1 + cot²x ) + ( 1 + tan²x ) .

    [ ∵ sin²x + cos²x = 1, cosec²x = 1 + cot²x  and  sec²x = 1 + tan²x ] .

= 1 + 4 + 1 + coy²x + 1 + tan²x .

= ( 7 + tan²x + cot²x ) = RHS .

∴ LHS = RHS .

Hence, it is proved .

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