Question

Prove that sin^2 35+sin^2 55-tan^2 45=0

Answer 1

The equation $\bf {sin}^{2} {35}^{ \circ} + {sin}^{2} {55}^{ \circ} - {tan}^{2} {45}^{ \circ} = 0$ has been proved.

Given:

• ${sin}^{2} {35}^{ \circ} + {sin}^{2} {55}^{ \circ} - {tan}^{2} {45}^{ \circ} = 0$

To find:

• Prove the equation.

Solution:

Concept to be used:

1. ${sin}^{2} { \theta} + {cos}^{2} { \theta} = 1$
2. ${sin} ({ {90}^{ \circ} - \theta}) = {cos} ({ \theta})$
3. $tan \: {45}^{ \circ} = 1$

Step 1:

Simplify the expression.

Take LHS;

Apply identity 2 in second term.

${sin}^{2} {35}^{ \circ} + {sin}^{2} ( {90}^{ \circ} - {35}^{ \circ} )- {tan}^{2} {45}^{ \circ}$

$= {sin}^{2} {35}^{ \circ} + {cos}^{2} ({35}^{ \circ} )- {tan}^{2} {45}^{ \circ}$

Step 2:

Apply identity.

Apply identity 1 and 3 in the expression.

$= {sin}^{2} {35}^{ \circ} + {cos}^{2} ({35}^{ \circ} )- {tan}^{2} {45}^{ \circ}$

$= 1 - 1$

$= 0$

LHS=RHS

Thus,

The trigonometric equation has been proved.

Learn more:

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tan 5 fan 25 tan 45 tan65 tan 85

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Answer 2

The identity that you are trying to prove is actually known as the Pythagorean Identity, which states that $cos^{2}$θ + $cos^{2}$θ = 1 for any angle θ. We can use this identity to prove the relationship between the sine and cosine functions.

Using the Pythagorean Identity, we can write:

$sin^{2}$35 + $cos^{2}$ 35 = 1

$sin^{2}$ 55 + $cos^{2}$ 55 = 1

Now, we can use the relationship between sine and tangent:

tan θ = sin θ / cos θ.

Substituting this into our equation, we get:

$tan^{2}$45 = $sin^{2}$ 45 / $cos^{2}$ 45

= 1 / $cos^{2}$ 45 = 1 / (1 - $sin^{2}$ 45)

Combining all the equations, we get:

$sin^{2}$ 35 + $sin^{2}$ 55 - $tan^{2}$ 45

= 1 - $cos^{2}$ 35 + 1 - $cos^{2}$ 55 - 1 / (1 - $sin^{2}$ 45)

= 2 -  $cos^{2}$ 35 -  $cos^{2}$ 55 - 1 / (1 - $sin^{2}$ 45)

= 2 -  $cos^{2}$  35 -  $cos^{2}$ 55 - 1 / (1 - $sin^{2}$45)

Now, we can use the Pythagorean Identity again to simplify the expression:

2 - $cos^{2}$ 35 - $cos^{2}$ 55 - 1 / (1 - $sin^{2}$45)

= 2 - $cos^{2}$ 35 -  $cos^{2}$ 55 - 1 / (1 - $sin^{2}$45

= 2 - $cos^{2}$ 35 -  $cos^{2}$ 55 - 1 / (1 - (1 / √2)^2)

= 2 -  $cos^{2}$ 35 - $cos^{2}$ 55 - 1 / (1 - 1 / 2)

= 2 - $cos^{2}$ 35 - $cos^{2}$ 55 - 2

= 0

Therefore,$sin^{2}$ 35 +$sin^{2}$ 55 - $tan^{2}$ 45 = 0.

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