Question

If tanⲪ +sinⲪ = m and

tanⲪ -sinⲪ = n ,

Show that -
${m}^{2} \: - {n}^{2} \: = 4 \sqrt{mn}$

☆CONTENT QUALITY SUPPORT☆

Answer 1

Hola Aiswary ,

tan ∅ + sin ∅ = m

tan ∅ - sin ∅ = n

To prove :

m² - n² = 4√mn

Proof :

LHS :

( tan ∅ + sin ∅ )² - ( tan ∅ - sin ∅ ) ²

= tan²∅ + sin²∅ + 2 tan ∅ × sin ∅ - [ tan ²∅ + sin ²∅ - 2 tan ∅ × sin ∅]

= tan ²∅ + sin²∅ + 2 tan ∅ × sin ∅ - tan²∅ - sin²∅ + 2 tan ∅ × sin ∅

= 4 tan ∅ × sin ∅

= 4 ( sin∅ / cos ∅ ) × sin ∅

= 4 sin²∅ / cos ∅........

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RHS :

4√mn

= 4 √ [ ( tan ∅ + sin ∅) ( tan ∅ - sin ∅ ) ]

= 4 √ [ tan² ∅ - sin² ∅ ]

= 4 √ [ ( sin²∅/cos²∅ ) - sin²∅ ]

= 4 √ [ ( sin²∅ - sin²∅ × cos²∅ ) / cos² ∅ ]

= 4 √ [ ( sin²∅ × sin²∅) / cos²∅ ]

= 4 × sin²∅ / cos∅..........

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Therefore LHS = RHS

Hope it helps.....

Answer 2

Heya friends ☺

Here is ur answer ...

-------------------------------
From LHS...

m²-n²

=(tan¢+sin¢)²-(tan¢-sin¢)²

=4tan¢×sin¢

➡Using this formula (a²+b²)-(a²-b²)=4ab


now from RHS =4√mn

=4√mn

=4×√(tan¢+sin¢)×(tan¢-sin¢)

=4×√tan²¢-sin²¢

=4×√sin²¢-sin²×cos²¢
------------------------------
cos¢

=4×sin¢/cos¢√(1-cos²¢)

=4×tan¢√sin²¢

=4tan¢×sin¢

Thus, here prooved ...

LHS =RHS

hence ,m²-n²=4√mn..

Hope it helps you..

#Rajukumar@@