Question

if tan A=n tan B and sinA=m sin B, prove that cos²A=m²-1/n²-1​

Answer 1

We have to find cos

2

A in terms of m and n. This means that angle B is to be eliminated from the given relations.

Now,

tanA=n tanB ⇒ tanB=

n

1

tanA ⇒ cotB=

tanA

n

and

sinA=msinB ⇒ sinB =

m

1

sinA ⇒ cosecB =

sinA

m

Substituting the values of cotB and cosecB in cosec

2

B−cot

2

B=1, we get,

⇒

sin

2

A

m

2

−

tan

2

A

n

2

=1

⇒

sin

2

A

m

2

−

sin

2

A

n

2

cos

2

A

=1

⇒

sin

2

A

m

2

−n

2

cos

2

A

=1

⇒m

2

−n

2

cos

2

A=sin

2

A

⇒m

2

−n

2

cos

2

A=1−cos

2

A

⇒m

2

−1=n

2

cos

2

A−cos

2

A

⇒m

2

−1=(n

2

−1)cos

2

A

⇒

n

2

−1

m

2

−1

=cos

2

A

Answer 2

Answer:

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